Linear K-Power Preservers and Trace of Power-Product Preservers

Abstract

Let V be the set of n×n complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix k∈Z\\01. We characterize linear maps ψ:V→V that satisfy ψAk=ψAk on an open neighborhood S of In in V. The k-power preservers are necessarily k-potent preservers, and k=2 corresponds to Jordan homomorphisms. Applying the results, we characterize maps ϕ,ψ:V→V that satisfy “trϕAψBk=trABk for all A∈V, B∈S, and ψ is linear” or “trϕAψBk=trABk for all A,B∈S and both ϕ and ψ are linear.” The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.

1. Introduction

Preserver problem is one of the most active research areas in matrix theory (e.g. [1, 2, 3, 4]). Researchers would like to characterize the maps on a given space of matrices preserving certain subsets, functions or relations. One of the preserver problems concerns maps $\psi$ on some sets $V$ of matrices which preserves $k$-power for a fixed integer $k\ge 2$, that is, $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for any $A\in V$ (e.g. [3, 5, 6]). The $k$-power preservers form a special class of polynomial preservers. One important reason of this problem lies on the fact that the case $k=2$ corresponds to Jordan homomorphisms. Moreover, every $k$-power preserver is also a $k$-potent preserver, that is, $A^k=A$ imply that $\psi {\left(A\right)}^k=\psi \left(A\right)$ for any $A\in V$. Some researches on $k$-potent preservers can be found in [6, 7, 8].

Given a field $\mathbb{F}$, let ${\mathcal{M}}_n\left(\mathbb{F}\right)$, ${\mathcal{S}}_n\left(\mathbb{F}\right)$, ${\mathcal{D}}_n\left(\mathbb{F}\right)$, ${\mathcal{N}}_n\left(\mathbb{F}\right)$, and ${\mathcal{T}}_n\left(\mathbb{F}\right)$ denote the set of $n\times n$ general, symmetric, diagonal, strictly upper triangular, and upper triangular matrices over $\mathbb{F}$, respectively. When $\mathbb{F}$ is the complex field $\mathbb{C}$, we may write ${\mathcal{M}}_n$ instead of ${\mathcal{M}}_n\left(\mathbb{C}\right)$, and so on. Let ${\mathcal{H}}_n$, ${\mathcal{P}}_n$, and $\overline{{\mathcal{P}}_n}$ denote the set of complex Hermitian, positive definite, and positive semidefinite matrices, and ${\mathcal{H}}_n\left(\mathbb{R}\right)={\mathcal{S}}_n\left(\mathbb{R}\right)$, ${\mathcal{P}}_n\left(\mathbb{R}\right)$, and $\overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}$ the corresponding set of real matrices, respectively. A matrix space is a subspace of ${\mathcal{M}}_{m,n}\left(\mathbb{F}\right)$ for certain $m,n\in {\mathbb{Z}}_{+}$. Let $A^t$ (resp. $A^{\ast }$) denote the transpose (resp. conjugate transpose) of a matrix $A$.

In 1951, Kadison [9] showed that a Jordan $\ast$-isomorphism on ${\mathcal{M}}_n$, namely, a bijective linear map with $\psi \left(A^2\right)=\psi {\left(A\right)}^2$ and $\psi \left(A^{\ast }\right)=\psi {\left(A\right)}^{\ast }$ for all $A\in {\mathcal{M}}_n$, is the direct sum of a $\ast$-isomorphism and a $\ast$-anti-isomorphism. Hence $\psi \left(A\right)={\mathit{UAU}}^{\ast }$ for all $A\in {\mathcal{M}}_n$ or $\psi \left(A\right)={\mathit{UA}}^T{U}^{\ast }$ for all $A\in {\mathcal{M}}_n$ by [[3], Theorem A.8]. Let $k\ge 2$ be a fixed integer. In 1992, Chan and Lim ([5]) determined a nonzero linear operator $\psi :{\mathcal{M}}_n\left(\mathbb{F}\right)\to {\mathcal{M}}_n\left(\mathbb{F}\right)$ (resp. $\psi :{\mathcal{S}}_n\left(\mathbb{F}\right)\to {\mathcal{S}}_n\left(\mathbb{F}\right)$) such that $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ (resp. ${\mathcal{S}}_n\left(\mathbb{F}\right)$) (See Theorems 3.1 and 5.1). In 1998, Brešar, Martindale, and Miers considered additive maps of general prime rings to solve an analogous problem by using the deep algebraic techniques ([10]). Monlár [[3], P6] described a particular case of their result which extends Theorem 3.1 to surjective linear operators on $\mathcal{B}\left(\mathcal{H}\right)$. In 2004, Cao and Zhang determined additive $k$-power preserver on ${\mathcal{M}}_n\left(\mathbb{F}\right)$ and ${\mathcal{S}}_n\left(\mathbb{F}\right)$ ([11]). They also characterized injective additive $k$-power preserver on ${\mathcal{T}}_n\left(\mathbb{F}\right)$ ([12] or [[6], Theorem 6.5.2]), which leads to injective linear $k$-power preserver on ${\mathcal{T}}_n\left(\mathbb{F}\right)$ (see Theorem 8.1). In 2006, Cao and Zhang also characterized linear $k$-power preservers from ${\mathcal{M}}_n\left(\mathbb{F}\right)$ to ${\mathcal{M}}_m\left(\mathbb{F}\right)$ and from ${\mathcal{S}}_n\left(\mathbb{F}\right)$ to ${\mathcal{M}}_m\left(\mathbb{F}\right)$ (resp. ${\mathcal{S}}_m\left(\mathbb{F}\right)$) [8].

In this article, given an integer $k\in \mathbb{Z}\backslash \left\{0,1\right\}$, we show that a unital linear map $\psi :V\to W$ between matrix spaces preserving $k$-powers on a neighborhood of identity must preserve all integer powers (Theorem 2.1). Then we characterize, for $\mathbb{F}=\mathbb{C}$ and $\mathbb{R}$, linear operators on sets $V={\mathcal{M}}_n\left(\mathbb{F}\right)$, ${\mathcal{H}}_n$, ${\mathcal{S}}_n\left(\mathbb{F}\right)$, ${\mathcal{P}}_n$, ${\mathcal{P}}_n\left(\mathbb{R}\right)$, ${\mathcal{D}}_n\left(\mathbb{F}\right)$, and ${\mathcal{T}}_n\left(\mathbb{F}\right)$ that satisfy $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ on an open neighborhood of $I_n$ in $V$. In the following descriptions, $P\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ is invertible, $U\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ is unitary, $O\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ is orthogonal, and $\lambda \in \mathbb{F}$ satisfies that ${\lambda }^{k-1}=1$.

1. $V={\mathcal{M}}_n\left(\mathbb{F}\right)$ (Theorem 3.4): $\psi \left(A\right)=\lambda {\mathit{PAP}}^{-1}$ or $\psi \left(A\right)=\lambda {\mathit{PA}}^t{P}^{-1}$.

2. $V={\mathcal{H}}_n$ (Theorem 4.1): When $k$ is even, $\psi \left(A\right)=U^{\ast }\mathit{AU}$ or $\psi \left(A\right)=U^{\ast }{A}^{t}U$. When $k$ is odd, $\psi \left(A\right)=\pm U^{\ast }\mathit{AU}$ or $\psi \left(A\right)=\pm U^{\ast }{A}^{t}U$.

3. $V={\mathcal{S}}_n\left(\mathbb{F}\right)$ (Theorem 5.2): $\psi \left(A\right)=\lambda {\mathit{OAO}}^t$.

4. $V={\mathcal{P}}_n$ or ${\mathcal{P}}_n\left(\mathbb{R}\right)$ (Theorem 6.1): $\psi \left(A\right)=U^{\ast }\mathit{AU}$ or $\psi \left(A\right)=U^{\ast }{A}^{t}U$.

5. $V={\mathcal{D}}_n\left(\mathbb{F}\right)$ (Theorem 7.1): $\psi \left(A\right)=\psi \left(I_n\right)diag\left(f_{p\left(1\right)}\left(A\right),\dots ,f_{p\left(n\right)}\left(A\right)\right)$, in which $\psi {\left(I_n\right)}^k=\psi \left(I_n\right)$, $p:\left\{1,\dots ,n\right\}\to \left\{0,1,\dots ,n\right\}$ is a function, and $f_i:{\mathcal{D}}_n\left(\mathbb{F}\right)\to \mathbb{F}$ $\left(i=0,1,\dots ,n\right)$ satisfy that, for $A=diag\left(a_1,\dots ,a_n\right)$, $f_0\left(A\right)=0$ and $f_i\left(A\right)=a_i$ for $i=1,\dots ,n$.

6. $V={\mathcal{T}}_n\left(\mathbb{F}\right)$ (Theorem 8.4 for $n\ge 3$): $\psi \left(A\right)=\lambda {\mathit{PAP}}^{-1}$ or $\psi \left(A\right)=\lambda {\mathit{PA}}^{-}{P}^{-1}$, in which $P\in {\mathcal{T}}_n\left(\mathbb{F}\right)$ and $A^{-}=\left(a_{n+1-j,n+1-i}\right)$ if $A=\left(a_{\mathit{ij}}\right)$.

Our results on ${\mathcal{M}}_n\left(\mathbb{F}\right)$ and ${\mathcal{S}}_n\left(\mathbb{F}\right)$ extend Chan and Lim’s results in Theorems 3.1 and 5.1, and result on ${\mathcal{T}}_n\left(\mathbb{F}\right)$ extend Cao and Zhang’s linear version result in [12].

Another topic is the study of a linear map $\varphi$ from a matrix set $S$ to another matrix set $T$ preserving trace equation. In 1931, Wigner’s unitary-antiunitary theorem [[3], p. 12] says that if $\varphi$ is a bijective map defined on the set of all rank one projections on a Hilbert space $H$ satisfying

$$\mathrm{tr}\left(\varphi \left(A\right)\varphi \left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right),$$E1

then there is an either unitary or antiunitary operator $U$ on $H$ such that $\varphi \left(P\right)=U^{\ast }\mathit{PU}$ or $\varphi \left(P\right)=U^{\ast }{P}^{t}U$ for all rank one projections $P$. In 1963, Uhlhorn generalized Wigner’s theorem to show that the same conclusion holds if the equality $\mathrm{tr}\left(\varphi \left(P\right)\varphi \left(Q\right)\right)=\mathrm{tr}\left(\mathit{PQ}\right)$ is replaced by $\mathrm{tr}\left(\varphi \left(P\right)\varphi \left(Q\right)\right)=0\iff \mathrm{tr}\left(\mathit{PQ}\right)=0$ (see [13]).

In 2002, Molnár (in the proof of [[14], Theorem 1]) showed that maps $\varphi$ on the space of all bounded linear operators on a Banach space $B\left(X\right)$ satisfying (1) for $A\in B\left(X\right)$, rank one operator $B\in B\left(X\right)$ are linear. In 2012, Li, Plevnik, and Šemrl [15] characterized bijective maps $\varphi :S\to S$ satisfying $\mathrm{tr}\left(\varphi \left(A\right)\varphi \left(B\right)\right)=c\iff \mathrm{tr}\left(\mathit{AB}\right)=c$ for a given real number $c$, where $S$ is ${\mathcal{H}}_n$, ${\mathcal{S}}_n\left(\mathbb{R}\right)$, or the set of rank one projections.

In [[16], Lemma 3.6], Huang et al. showed that the following statements are equivalent for a unital map $\varphi$ on ${\mathcal{P}}_n$:

1. $\mathit{tr}\left(\varphi \left(A\right)\varphi \left(B\right)\right)=\mathit{tr}\left(\mathit{AB}\right)$ for $A,B\in {\mathcal{P}}_n$;

2. $\mathit{tr}\left(\varphi \left(A\right)\varphi {\left(B\right)}^{-1}\right)=\mathit{tr}\left({\mathit{AB}}^{-1}\right)$ for $A,B\in {\mathcal{P}}_n$;

3. $\varphi \left(A\right)=U^{\ast }\mathit{AU}$ or $U^{\ast }{A}^{t}U$ for a unitary matrix $U$.

The authors also determined the cases if $\varphi$ is not assuming unital, the set ${\mathcal{P}}_n$ is replaced by another set like ${\mathcal{M}}_n$, ${\mathcal{S}}_n$, ${\mathcal{T}}_n$, or ${\mathcal{D}}_n$. In [[17], Theorem 3.8], Leung, Ng, and Wong considered the relation (1) on infinite dimensional space.

Let $⟨S⟩$ denote the subspace spanned by a subset $S$ of a vector space. Recently, Huang and Tsai studied two maps preserving trace of product [18]. Suppose two maps $\varphi :V_1\to W_1$ and $\psi :V_2\to W_2$ between subsets of matrix spaces over a field $\mathbb{F}$ under some conditions satisfy

$$\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right)$$E2

for all $A\in V_1$, $B\in V_2$. The authors showed that these two maps can be extended to bijective linear maps $\tilde{\varphi }:⟨V_1⟩\to ⟨W_1⟩$ and $\tilde{\psi }:⟨V_2⟩\to ⟨W_2⟩$ that satisfy $\mathrm{tr}\left(\tilde{\varphi }\left(A\right)\tilde{\psi }\left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right)$ for all $A\in ⟨V_1⟩$, $B\in ⟨V_2⟩$ (see Theorem 2.2). Hence when a matrix space $V$ is closed under conjugate transpose, every linear bijection $\varphi :V\to V$ corresponds to a unique linear bijection $\psi :V\to V$ that makes (2) hold (see Corollary 2.3). Therefore, each of $\varphi$ and $\psi$ has no specific form.

One natural question is to ask when the following equality holds for a fixed $k\in \mathbb{Z}\backslash \left\{0,1\right\}$:

$$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left({\mathit{AB}}^k\right).$$E3

The second major work of this paper is to use our descriptions of linear $k$-power preservers on an open neighborhood $S$ of $I_n$ in $V$ to characterize maps $\varphi ,\psi :V\to V$ under one of the assumptions:

1. equality (3) holds for all $A\in V$, $B\in S$, and $\psi$ is linear, or

2. equality (3) holds for all $A,B\in S$ and both $\varphi$ and $\psi$ are linear,

for the sets $V={\mathcal{M}}_n$, ${\mathcal{H}}_n$, ${\mathcal{P}}_n$, ${\mathcal{S}}_n$, ${\mathcal{D}}_n$, ${\mathcal{T}}_n$, and their real counterparts. These results, together with Theorem 2.2 and the characterizations of maps ${\varphi }_1,\cdots ,{\varphi }_m:V\to V$ ($m\ge 3$) that satisfy $\mathit{tr}\left({\varphi }_1\left(A_1\right)\cdots {\varphi }_m\left(A_m\right)\right)=\mathit{tr}\left(A_1\cdots A_m\right)$ in [18], make a comprehensive picture of the preservers of trace of matrix products in the related matrix spaces and sets.

In the following characterizations, $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$, $P,Q\in M_n\left(\mathbb{F}\right)$ are invertible, $U\in M_n\left(\mathbb{F}\right)$ is unitary, $O\in M_n\left(\mathbb{F}\right)$ is orthogonal, and $c\in \mathbb{F}\backslash \left\{0\right\}$.

1. $V={\mathcal{M}}_n\left(\mathbb{F}\right)$ (Theorem 3.5):

   1. When $k=-1$, $\varphi \left(A\right)=\mathit{PAQ}$ and $\psi \left(B\right)=\mathit{PBQ}$, or $\varphi \left(A\right)={\mathit{PA}}^{t}Q$ and $\psi \left(B\right)={\mathit{PB}}^{t}Q$.

   2. When $k\in \mathbb{Z}\backslash \left\{-\mathrm{1,0,1}\right\}$, $\varphi \left(A\right)=c^{-k}{\mathit{PAP}}^{-1}$ and $\psi \left(B\right)={\mathit{\text{cPBP}}}^{-1}$, or $\varphi \left(A\right)=c^{-k}{\mathit{PA}}^t{P}^{-1}$ and $\psi \left(B\right)={\mathit{cPB}}^t{P}^{-1}$.

2. $V={\mathcal{H}}_n$ (Theorem 4.2):

   1. When $k=-1$, $\varphi \left(A\right)={\mathit{cP}}^{\ast }\mathit{AP}$ and $\psi \left(B\right)={\mathit{cP}}^{\ast }\mathit{BP}$, or $\varphi \left(A\right)={\mathit{cP}}^{\ast }{A}^{t}P$ and $\psi \left(B\right)={\mathit{cP}}^{\ast }{B}^{t}P$, for $c\in \left\{1,-1\right\}$.

   2. When $k\in \mathbb{Z}\backslash \left\{-\mathrm{1,0,1}\right\}$, $\varphi \left(A\right)=c^{-k}{U}^{\ast }\mathit{AU}$ and $\psi \left(B\right)={\mathit{cU}}^{\ast }\mathit{BU}$, or $\varphi \left(A\right)=c^{-k}{U}^{\ast }{A}^{t}U$ and $\psi \left(B\right)={\mathit{cU}}^{\ast }{B}^{t}U$, for $c\in \mathbb{R}\backslash \left\{0\right\}$.

3. $V={\mathcal{S}}_n\left(\mathbb{F}\right)$ (Theorem 5.3):

   1. When $k=-1$, $\varphi \left(A\right)={\mathit{\text{cPAP}}}^t$ and $\psi \left(B\right)={\mathit{\text{cPBP}}}^t$.

   2. When $k\in \mathbb{Z}\backslash \left\{-\mathrm{1,0,1}\right\}$, $\varphi \left(A\right)=c^{-k}{\mathit{OAO}}^t$ and $\psi \left(B\right)={\mathit{\text{cOBO}}}^t$.

4. $V={\mathcal{P}}_n$ and ${\mathcal{P}}_n\left(\mathbb{R}\right)$ (Theorem 6.4): $\varphi \left(A\right)=c^{-k}{U}^{\ast }\mathit{AU}$ and $\psi \left(B\right)={\mathit{cU}}^{\ast }\mathit{BU}$, or $\varphi \left(A\right)=c^{-k}{U}^{\ast }{A}^{t}U$ and $\psi \left(B\right)={\mathit{cU}}^{\ast }{B}^{t}U$, in which $c\in {\mathbb{R}}^{+}$. Characterizations under some other assumptions are also given as special cases of Theorem 6.2 (Huang, Tsai [18]).

5. $V={\mathcal{D}}_n\left(\mathbb{F}\right)$ (Theorem 7.2): $\varphi \left(A\right)={\mathit{PC}}^{-k}{\mathit{AP}}^{-1},\psi \left(B\right)={\mathit{\text{PCBP}}}^{-1}$ where $P$ is a permutation matrix and $C={\mathcal{D}}_n\left(\mathbb{F}\right)$ is diagonal and invertible.

6. $V={\mathcal{T}}_n\left(\mathbb{F}\right)$ (Theorem 8.5): $\varphi$ and $\psi$ send ${\mathcal{N}}_n\left(\mathbb{F}\right)$ to ${\mathcal{N}}_n\left(\mathbb{F}\right)$, ${\left.\left(D\circ \varphi \right)\right|}_{{\mathcal{D}}_n\left(\mathbb{F}\right)}$ and ${\left.\left(D\circ \psi \right)\right|}_{{\mathcal{D}}_n\left(\mathbb{F}\right)}$ are characterized by Theorem 7.2, and $D\circ \varphi =D\circ \varphi \circ D$. Here $D$ denotes the map that sends $A\in {\mathcal{T}}_n\left(\mathbb{F}\right)$ to the diagonal matrix with the same diagonal as $A$.

The sets ${\mathcal{M}}_n$, ${\mathcal{H}}_n$, ${\mathcal{P}}_n$, ${\mathcal{S}}_n,$ ${\mathcal{D}}_n$, and their real counterparts are closed under conjugate transpose. In these sets, $\mathit{tr}\left(\mathit{AB}\right)=⟨A^{\ast },B⟩$ for the standard inner product. Our trace of product preservers can also be interpreted as inner product preservers, which have wide applications in research areas like quantum information theory.

2. Preliminary

2.1 Linear operators preserving powers

We show below that: given $k\in \mathbb{Z}\backslash \left\{0,1\right\}$, a unital linear map $\psi :V\to W$ between matrix spaces preserving $k$-powers on a neighborhood of identity in $V$ must preserve all integer powers. Let ${\mathbb{Z}}_{+}$ (resp. ${\mathbb{Z}}_{-}$) denote the set of all positive (resp. negative) integers.

Theorem 2.1. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let $V\subseteq {\mathcal{M}}_p\left(\mathbb{F}\right)$ and $W\subseteq {\mathcal{M}}_q\left(\mathbb{F}\right)$ be matrix spaces. Fix $k\in \mathbb{Z}\backslash \left\{0,1\right\}$.

1. Suppose the identity matrix $I_p\in V$ and $A^k\in V$ for all matrices $A$ in an open neighborhood ${\mathcal{S}}_V$ of $I_p$ in $V$ consisting of invertible matrices. Then

$$\left\{\mathit{AB}+\mathit{BA}:A,B\in V\right\}\subseteq V,$$E4

$$\left\{A^{-1}:A\in V\mathit{\text{is invertible}}\right\}\subseteq V.$$E5

In particular,

$$\left\{A^r:A\in V\right\}\subseteq V,r\in {\mathbb{Z}}_{+},\mathit{\text{and}}$$E6

$$\left\{A^r:A\in V\mathit{\text{is invertible}}\right\}\subseteq V,r\in {\mathbb{Z}}_{-}.$$E7

1. 2. Suppose $I_p\in V$, $I_q\in W$, and $A^k\in V$ for all matrices $A$ in an open neighborhood ${\mathcal{S}}_V$ of $I_p$ in $V$ consisting of invertible matrices. Suppose $\psi :V\to W$ is a linear map that satisfies the following conditions:

$$\psi \left(I_p\right)=I_q,$$E8

$$\psi \left(A^k\right)=\psi {\left(A\right)}^k,A\in {\mathcal{S}}_V.$$E9

Then

$$\psi \left(\mathit{AB}+\mathit{BA}\right)=\psi \left(A\right)\psi \left(B\right)+\psi \left(B\right)\psi \left(A\right),A,B\in V,$$E10

$$\psi \left(A^{-1}\right)=\psi {\left(A\right)}^{-1},\mathit{\text{invertible}}A\in V.$$E11

In particular,

$$\psi \left(A^r\right)=\psi {\left(A\right)}^r,A\in V,r\in {\mathbb{Z}}_{+},\mathit{\text{and}}$$E12

$$\psi \left(A^r\right)=\psi {\left(A\right)}^r,\mathit{\text{invertible}}A\in V,r\in {\mathbb{Z}}_{-}.$$E13

Proof. We prove the complex case. The real case is done similarly.

1. 1. For each $A\in V\backslash \left\{0\right\}$, there is $\mathrm{ϵ}>0$ such that $I_p+\mathit{xA}\in {\mathcal{S}}_V$ for all $x\in \mathbb{C}$ with $\mid x\mid <min\left\{\mathrm{ϵ},\frac{1}{\parallel A\parallel }\right\}$. Thus

$${\left(I_p+\mathit{xA}\right)}^k=I_p+\mathit{xkA}+x^2\frac{k\left(k-1\right)}{2}{A}^2+\cdots \in V.$$E14

The second derivative

$${\left.\frac{d^2}{{\mathit{dx}}^2}{\left(I_p+\mathit{xA}\right)}^k\right|}_{x=0}=k\left(k-1\right)A^2\in V.$$E15

Since $k\cancel{\in }\left\{0,1\right\}$, we have $A^2\in V$ for all $A\in V$. Therefore, for $A,B\in V$,

$$\mathit{AB}+\mathit{BA}={\left(A+B\right)}^2-A^2-B^2\in V.$$E16

In particular, $A\in V$ implies that $A^r\in V$ for all $r\in {\mathbb{Z}}_{+}$.

Cayley-Hamilton theorem implies that every invertible matrix $A$ satisfies that $A^{-1}=f\left(A\right)$ for a certain polynomial $f\left(x\right)\in \mathbb{F}\left[x\right]$. Therefore, $A^{-1}\in V$, so that $A^r\in V$ for all $r\in {\mathbb{Z}}_{-}$.

1. 2. Now suppose (8) and (9) hold. The proof is proceeded similarly to the proof of part (1). For every $A\in V$, there is $\mathrm{ϵ}>0$ such that for all $x\in \mathbb{C}$ with $\mid x\mid <min\left\{\mathrm{ϵ},\frac{1}{\parallel A\parallel },\frac{1}{\parallel \psi \left(A\right)\parallel }\right\}$,

$${\left(\psi \left(I_p+\mathit{xA}\right)\right)}^k=I_q+\mathit{xk\psi }\left(A\right)+x^2\frac{k\left(k-1\right)}{2}\psi {\left(A\right)}^2+\cdots \in W,$$E17

$$\psi \left({\left(I_p+\mathit{xA}\right)}^k\right)=I_q+\mathit{xk\psi }\left(A\right)+x^2\frac{k\left(k-1\right)}{2}\psi \left(A^2\right)+\cdots \in W.$$E18

since (17) and (18) equal, we have

$$\psi {\left(A\right)}^2=\psi \left(A^2\right),A\in V.$$E19

Therefore, for $A,B\in V$,

$$\psi \left({\left(A+B\right)}^2\right)=\psi {\left(A+B\right)}^2$$E20

We get (10): $\psi \left(\mathit{AB}+\mathit{BA}\right)=\psi \left(A\right)\psi \left(B\right)+\psi \left(B\right)\psi \left(A\right)$. In particular $\psi \left(A^r\right)=\psi {\left(A\right)}^r$ for all $A\in V$ and $r\in {\mathbb{Z}}_{+}$.

Every invertible $A\in V$ can be expressed as $A^{-1}=f\left(A\right)$ for a certain polynomial $f\left(x\right)\in \mathbb{F}\left[x\right]$. Then $\psi \left(A^{-1}\right)=\psi \left(f\left(A\right)\right)=f\left(\psi \left(A\right)\right)$ is commuting with $\psi \left(A\right)$. Hence

$$2\psi \left(A^{-1}\right)\psi \left(A\right)=\psi \left(A^{-1}\right)\psi \left(A\right)+\psi \left(A\right)\psi \left(A^{-1}\right)=\psi \left(A^{-1}A+{\mathit{AA}}^{-1}\right)=2I_q.$$E21

We get $\psi \left(A^{-1}\right)=\psi {\left(A\right)}^{-1}$. Therefore, $\psi \left(A^r\right)=\psi {\left(A\right)}^r$ for all $r\in {\mathbb{Z}}_{-}$.

Theorem 2.1 is powerful in exploring $k$-power preservers in matrix spaces. Note that every $k$-power preserver is a $k$-potent preserver. Theorem 2.1 can also be used to investigate $k$-potent preservers in matrix spaces.

2.2 Two maps preserving trace of product

We recall two results about two maps preserving trace of product in [18]. They are handy in proving linear bijectivity of maps preserving trace of products. Recall that if $S$ is a subset of a vector space, then $⟨S⟩$ denotes the subspace spanned by $S$.

Theorem 2.2 (Huang, Tsai [18]). Let $\varphi :V_1\to W_1$ and $\psi :V_2\to W_2$ be two maps between subsets of matrix spaces over a field $\mathbb{F}$ such that:

$$dim⟨V_1⟩=dim⟨V_2⟩\ge max\left\{dim⟨W_1⟩,dim⟨W_2⟩\right\}.$$

1. $\mathit{AB}$ are well-defined square matrices for $\left(A,B\right)\in \left(V_1\times V_2\right)\cup \left(W_1\times W_2\right)$.

2. If $A\in ⟨V_1⟩$ satisfies that $\mathrm{tr}\left(\mathit{AB}\right)=0$ for all $B\in ⟨V_2⟩$, then $A=0$.

3. $\varphi$ and $\psi$ satisfy that

$$\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right),A\in V_1,B\in V_2.$$E22

Then $dim⟨V_1⟩=dim⟨V_2⟩=dim⟨W_1⟩=dim⟨W_2⟩$ and $\varphi$ and $\psi$ can be extended to bijective linear map $\tilde{\varphi }:⟨V_1⟩\to ⟨W_1⟩$ and $\tilde{\psi }:⟨V_2⟩\to ⟨W_2⟩$, respectively, such that

$$\mathrm{tr}\left(\tilde{\varphi }\left(A\right)\tilde{\psi }\left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right),A\in ⟨V_1⟩,B\in ⟨V_2⟩.$$E23

A subset $V$ of ${\mathcal{M}}_n$ is closed under conjugate transpose if $\left\{A^{\ast }:A\in V\right\}\subseteq V$. A real or complex matrix space $V$ is closed under conjugate transpose if and only if $V$ equals the direct sum of its subspace of Hermitian matrices and its subspace of skew-Hermitian matrices.

Corollary 2.3 (Huang, Tsai [18]). Let $V$ be a subset of ${\mathcal{M}}_n$ closed under conjugate transpose. Suppose two maps $\varphi ,\psi :V\to V$ satisfy that

$$\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right),A,B\in V.$$E24

Then $\varphi$ and $\psi$ can be extended to linear bijections on $⟨V⟩$. Moreover, when $V$ is a vector space, every linear bijection $\varphi :V\to V$ corresponds to a unique linear bijection $\psi :V\to V$ such that (24) holds. Explicitly, given an orthonormal basis $\left\{A_1,\dots ,A_{\mathrm{\ell }}\right\}$ of $V$ with respect to the inner product $⟨A,B⟩=\mathrm{tr}\left(A^{\ast }B\right)$, $\psi$ is defined by $\psi \left(A_i\right)=B_i$ in which $\left\{B_1,\dots ,B_{\mathrm{\ell }}\right\}$ is a basis of $V$ with $\mathrm{tr}\left(\varphi \left(A_i^{\ast }\right)B_j\right)={\delta }_{i,j}$ for all $i,j\in \left\{1,\dots ,\mathrm{\ell }\right\}$.

Corollary 2.3 shows that when a matrix space $V$ is closed under conjugate transpose, every linear bijection $\varphi :V\to V$ corresponds to a unique linear bijection $\psi :V\to V$ that makes (24) hold. The next natural thing is to determine $\varphi$ and $\psi$ that satisfy $\mathit{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathit{tr}\left({\mathit{AB}}^k\right)$ for a fixed $k\in \mathbb{Z}\backslash \left\{0,1\right\}.$

From now on, we focus on the fields $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$.

3. $k$-power linear preservers and trace of power-product preservers on ${\mathcal{M}}_n$ and ${\mathcal{M}}_n\left(\mathbb{R}\right)$

3.1 $k$-power preservers on ${\mathcal{M}}_n$ and ${\mathcal{M}}_n\left(\mathbb{R}\right)$

Chan and Lim described the linear $k$-power preservers on ${\mathcal{M}}_n$ and ${\mathcal{M}}_n\left(\mathbb{R}\right)$ for $k\ge 2$ in [7, Theorem 1] as follows.

Theorem 3.1. (Chan, Lim [5]) Let an integer $k\ge 2$. Let $\mathbb{F}$ be a field with $\mathit{char}\left(\mathbb{F}\right)=0$ or $\mathit{char}\left(\mathbb{F}\right)>k$. Suppose that $\psi :{\mathcal{M}}_n\left(\mathbb{F}\right)\to {\mathcal{M}}_n\left(\mathbb{F}\right)$ is a nonzero linear operator such that $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A\in {\mathcal{M}}_n\left(\mathbb{F}\right)$. Then there exist $\lambda \in \mathbb{F}$ with ${\lambda }^{k-1}=1$ and an invertible matrix $P\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\psi \left(A\right)=\lambda {\mathit{PAP}}^{-1},A\in {\mathcal{M}}_n\left(\mathbb{F}\right),\mathit{\text{or}}$$E25

$$\psi \left(A\right)=\lambda {\mathit{PA}}^t{P}^{-1},A\in {\mathcal{M}}_n\left(\mathbb{F}\right).$$E26

(25) and (26) need not hold if $\psi$ is zero or is a map on a subspace of ${\mathcal{M}}_n\left(\mathbb{F}\right)$. The following are two examples. Another example can be found in maps on ${\mathcal{D}}_n\left(\mathbb{F}\right)$ (Theorem 7.1).

Example 3.2. The zero map $\psi \left(A\right)\equiv 0$ clearly satisfies $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A\in {\mathcal{M}}_n$ but they are not of the form (25) or (26).

Example 3.3. Let $n=k+m$, $k,m\ge 2$, and consider the operator $\psi$ on the subspace $W={\mathcal{M}}_k\oplus {\mathcal{M}}_m$ of ${\mathcal{M}}_n$ defined by $\psi \left(A\oplus B\right)=A\oplus B^t$ for $A\in {\mathcal{M}}_k$ and $B\in {\mathcal{M}}_m.$ Then $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A\in W$ and $k\in {\mathbb{Z}}_{+}$, but $\psi$ is not of the form (25) or (26).

We now generalize Theorem 3.1 to include negative integers $k$ and to assume the $k$-power preserving condition $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ only on matrices nearby the identity.

Theorem 3.4. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let an integer $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. Suppose that $\psi :{\mathcal{M}}_n\left(\mathbb{F}\right)\to {\mathcal{M}}_n\left(\mathbb{F}\right)$ is a nonzero linear map such that $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A$ in an open neighborhood of $I_n$ consisting of invertible matrices. Then there exist $\lambda \in \mathbb{F}$ with ${\lambda }^{k-1}=1$ and an invertible matrix $P\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\psi \left(A\right)=\lambda {\mathit{PAP}}^{-1},A\in {\mathcal{M}}_n\left(\mathbb{F}\right),\mathit{\text{or}}$$E27

$$\psi \left(A\right)=\lambda {\mathit{PA}}^t{P}^{-1},A\in {\mathcal{M}}_n\left(\mathbb{F}\right).$$E28

Proof. We prove for the case $\mathbb{F}=\mathbb{C}$. The case $\mathbb{F}=\mathbb{R}$ can be done similarly. Obviously, $\psi \left(I_n\right)=\psi \left(I_n^k\right)=\psi {\left(I_n\right)}^k$.

1. First suppose $k\ge 2$. For each $A\in {\mathcal{M}}_n$, there exists $\mathrm{ϵ}>0$ such that for all $x\in \mathbb{C}$ with $\mid x\mid <\mathrm{ϵ}$, the following two power series converge and equal:

$$\begin{array}{l}{\left(\psi \left(I_n+\mathit{xA}\right)\right)}^k=\psi \left(I_n\right)+x\left(\sum _{i=0}^{k-1}\psi {\left(I_n\right)}^i\psi \left(A\right)\psi {\left(I_n\right)}^{k-1-i}\right)\\ +x^2\left(\sum _{i=0}^{k-2}\sum _{j=0}^{k-2-i}\psi {\left(I_n\right)}^i\psi \left(A\right)\psi {\left(I_n\right)}^j\psi \left(A\right)\psi {\left(I_n\right)}^{k-2-i-j}\right)+\cdots \end{array}$$E29

$$\psi \left({\left(I_n+\mathit{xA}\right)}^k\right)=\psi \left(I_n\right)+\mathit{xk\psi }\left(A\right)+x^2\frac{k\left(k-1\right)}{2}\psi \left(A^2\right)+\cdots$$E30

Equating degree one terms above, we get

$$\mathit{k\psi }\left(A\right)=\sum _{i=0}^{k-1}\psi {\left(I_n\right)}^i\psi \left(A\right)\psi {\left(I_n\right)}^{k-1-i}.$$E31

Applying (31), we have

$$\mathit{k\psi }\left(I_n\right)\psi \left(A\right)-\mathit{k\psi }\left(A\right)\psi \left(I_n\right)=\psi {\left(I_n\right)}^k\psi \left(A\right)-\psi \left(A\right)\psi {\left(I_n\right)}^k=\psi \left(I_n\right)\psi \left(A\right)-\psi \left(A\right)\psi \left(I_n\right).$$E32

Hence $\psi \left(I_n\right)\psi \left(A\right)=\psi \left(A\right)\psi \left(I_n\right)$ for $A\in {\mathcal{M}}_n$, that is, $\psi \left(I_n\right)$ commutes with the range of $\psi$.

Now equating degree two terms of (29) and (30) and taking into account that $k\cancel{\in }\left\{0,1\right\}$, we have

$$\psi {\left(I_n\right)}^{k-2}\psi {\left(A\right)}^2=\psi \left(A^2\right).$$E33

Define ${\psi }_1\left(A\right)=\psi {\left(I_n\right)}^{k-2}\psi \left(A\right)$ for $A\in {\mathcal{M}}_n$. Then ${\psi }_1\left(A^2\right)={\left({\psi }_1\left(A\right)\right)}^2$ for all $A\in {\mathcal{M}}_n$. (31) and the assumption that $\psi$ is nonzero imply that $\psi \left(I_n\right)\ne 0$. So ${\psi }_1\left(I_n\right)\psi \left(I_n\right)=\psi {\left(I_n\right)}^k=\psi \left(I_n\right)\ne 0$. Thus ${\psi }_1\left(I_n\right)\ne 0$ and ${\psi }_1$ is nonzero. By Theorem 3.1, there exists an invertible $P\in {\mathcal{M}}_n$ such that ${\psi }_1\left(A\right)={\mathit{PAP}}^{-1}$ for $A\in {\mathcal{M}}_n$ or ${\psi }_1\left(A\right)={\mathit{PA}}^t{P}^{-1}$ for $A\in {\mathcal{M}}_n$. Moreover, $\psi \left(I_n\right)$ commutes with all ${\psi }_1\left(A\right)$, so that $\psi \left(I_n\right)=\lambda I_n$ for a $\lambda \in \mathbb{C}$. By $I_n={\psi }_1\left(I_n\right)=\psi {\left(I_n\right)}^{k-1}$, we get ${\lambda }^{k-1}=1$. Therefore, $\psi \left(A\right)={\mathit{\lambda \psi }}_1\left(A\right)$. We get (27) and (28).

1. Next Suppose $k<0$. For every $A\in {\mathcal{M}}_n$, the power series expansions of ${\left(\psi \left(I_n+\mathit{xA}\right)\right)}^{-k}$ and $\psi {\left({\left(I_n+\mathit{xA}\right)}^k\right)}^{-1}$ are equal when $\mid x\mid$ is sufficiently small:

$${\left(\psi \left(I_n+\mathit{xA}\right)\right)}^{-k}=\psi {\left(I_n\right)}^{-1}+x\left(\sum _{i=0}^{-k-1}\psi {\left(I_n\right)}^i\psi \left(A\right)\psi {\left(I_n\right)}^{-k-1-i}\right)+\cdots$$E34

$$\psi {\left({\left(I_n+\mathit{xA}\right)}^k\right)}^{-1}=\psi {\left(I_n\right)}^{-1}-\mathit{xk\psi }{\left(I_n\right)}^{-1}\psi \left(A\right)\psi {\left(I_n\right)}^{-1}+\cdots$$E35

Equating degree one terms of (34) and (35), we get

$$-\mathit{k\psi }{\left(I_n\right)}^{-1}\psi \left(A\right)\psi {\left(I_n\right)}^{-1}=\sum _{i=0}^{-k-1}\psi {\left(I_n\right)}^i\psi \left(A\right)\psi {\left(I_n\right)}^{-k-1-i}.$$E36

Therefore,

$$\begin{array}{l}-k\left(\psi \left(A\right)\psi {\left(I_n\right)}^{-1}-\psi {\left(I_n\right)}^{-1}\psi \left(A\right)\right)\\ =\psi \left(I_n\right)\left(\sum _{i=0}^{-k-1}\psi {\left(I_n\right)}^i\psi \left(A\right)\psi {\left(I_n\right)}^{-k-1-i}\right)-\left(\sum _{i=0}^{-k-1}\psi {\left(I_n\right)}^i\psi \left(A\right)\psi {\left(I_n\right)}^{-k-1-i}\right)\psi \left(I_n\right)\\ =\psi {\left(I_n\right)}^{-k}\psi \left(A\right)-\psi \left(A\right)\psi {\left(I_n\right)}^{-k}=\psi {\left(I_n\right)}^{-1}\psi \left(A\right)-\psi \left(A\right)\psi {\left(I_n\right)}^{-1}.\end{array}$$E37

We get $\psi {\left(I_n\right)}^{-1}\psi \left(A\right)=\psi \left(A\right)\psi {\left(I_n\right)}^{-1}$ for $A\in {\mathcal{M}}_n$. So $\psi {\left(I_n\right)}^{-1}$ and $\psi \left(I_n\right)$ commute with the range of $\psi$. The following power series are equal for every $A\in {\mathcal{M}}_n$ when $\mid x\mid$ is sufficiently small:

$${\left(\psi \left(I_n+\mathit{xA}\right)\right)}^k=\psi \left(I_n\right)+\mathit{xk\psi }{\left(I_n\right)}^{k-1}\psi \left(A\right)+x^2\frac{k\left(k-1\right)}{2}\psi {\left(I_n\right)}^{k-2}\psi {\left(A\right)}^2+\cdots$$E38

$$\psi \left({\left(I_n+\mathit{xA}\right)}^k\right)=\psi \left(I_n\right)+\mathit{xk\psi }\left(A\right)+x^2\frac{k\left(k-1\right)}{2}\psi \left(A^2\right)+\cdots$$E39

Equating degree two terms of (38) and (39), we get $\psi {\left(I_n\right)}^{k-2}\psi {\left(A\right)}^2=\psi \left(A^2\right).$ Let ${\psi }_1\left(A\right)≔\psi {\left(I_n\right)}^{k-2}\psi \left(A\right)=\psi {\left(I_n\right)}^{-1}\psi \left(A\right)$. Then ${\psi }_1{\left(A\right)}^2={\psi }_1\left(A^2\right)$ and ${\psi }_1$ is nonzero. Using Theorem 3.1, we can get (27) and (39).

3.2 Trace of power-product preserers on ${\mathcal{M}}_n$ and ${\mathcal{M}}_n\left(\mathbb{R}\right)$

Corollary 2.3 shows that every linear bijection $\varphi :{\mathcal{M}}_n\left(\mathbb{F}\right)\to {\mathcal{M}}_n\left(\mathbb{F}\right)$ corresponds to another linear bijection $\psi :{\mathcal{M}}_n\left(\mathbb{F}\right)\to {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that $\mathit{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathit{tr}\left(\mathit{AB}\right)$ for all $A,B\in {\mathcal{M}}_n\left(\mathbb{F}\right)$. When $m\ge 3$, maps ${\varphi }_1,\cdots ,{\varphi }_m$ on ${\mathcal{M}}_n\left(\mathbb{F}\right)$ that satisfy $\mathit{tr}\left({\varphi }_1\left(A_1\right)\cdots {\varphi }_m\left(A_m\right)\right)=\mathit{tr}\left(A_1\cdots A_m\right)$ for $A_1,\dots ,A_m\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ are determined in [18].

If two maps on ${\mathcal{M}}_n\left(\mathbb{F}\right)$ satisfy the following trace condition about $k$-powers, then they have specific forms.

Theorem 3.5. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. Let $S$ be an open neighborhood of $I_n$ consisting of invertible matrices. Then two maps $\varphi ,\psi :{\mathcal{M}}_n\left(\mathbb{F}\right)\to {\mathcal{M}}_n\left(\mathbb{F}\right)$ satisfy that

$$\mathit{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathit{tr}\left({\mathit{AB}}^k\right),$$E40

1. for all $A\in {\mathcal{M}}_n\left(\mathbb{F}\right),B\in S$, and $\psi$ is linear, or

2. for all $A,B\in S$ and both $\varphi$ and $\psi$ are linear,

if and only if $\varphi$ and $\psi$ take the following forms:

1. a. When $k=-1$, there exist invertible matrices $P,Q\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\left(\begin{array}{c}\varphi \left(A\right)=\mathit{PAQ}\\ \psi \left(B\right)=\mathit{PBQ}\end{array}\right.\mathit{\text{or}}\left(\begin{array}{c}\varphi \left(A\right)={\mathit{PA}}^{t}Q\\ \psi \left(B\right)={\mathit{PB}}^{t}Q\end{array}\right.A,B\in {\mathcal{M}}_n\left(\mathbb{F}\right).$$E41

1. b. When $k\in \mathbb{Z}\backslash \left\{-\mathrm{1,0,1}\right\}$, there exist $c\in \mathbb{F}\backslash \left\{0\right\}$ and an invertible matrix $P\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\left(\begin{array}{c}\varphi \left(A\right)=c^{-k}{\mathit{PAP}}^{-1}\\ \psi \left(B\right)={\mathit{\text{cPBP}}}^{-1}\end{array}\right.\mathit{\text{or}}\left(\begin{array}{c}\varphi \left(A\right)=c^{-k}{\mathit{PA}}^t{P}^{-1}\\ \psi \left(B\right)={\mathit{cPB}}^t{P}^{-1}\end{array}\right.A,B\in {\mathcal{M}}_n\left(\mathbb{F}\right).$$E42

Proof. We prove the case $\mathbb{F}=\mathbb{C}$; the case $\mathbb{F}=\mathbb{R}$ can be done similarly.

Suppose assumption (2) holds. Then for every $A\in {\mathcal{M}}_n\left(\mathbb{F}\right)$, there exists $c\in \mathbb{F}\backslash \left\{0\right\}$ such that $I_n-\mathit{cA}\in S$, so that for all $B\in S$:

$$\mathrm{tr}\left(B^k\right)=\mathrm{tr}\left(\left(\varphi \left(I_n-\mathit{cA}\right)+\mathit{c\varphi }\left(A\right)\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left(\left(I_n-\mathit{cA}\right)B^k\right)+c\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right).$$E43

Thus $\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left({\mathit{AB}}^k\right)$ for $A\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ and $B\in S$, which leads to assumption (1).

Now we prove the theorem under assumption (1), that is, (40) holds for all $A\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ and $B\in S$, and $\psi$ is linear. Only the necessary part is needed to prove.

Let $S^{\prime }=\left\{B\in \overline{{\mathcal{P}}_n}:B^{1/k}\in S\right\}$, which is an open neighborhood of $I_n$ in $\overline{{\mathcal{P}}_n}$. Define $\tilde{\psi }:S^{\prime }\to {\mathcal{M}}_n$ such that $\tilde{\psi }\left(B\right)=\psi {\left(B^{1/k}\right)}^k$. Then (40) implies that

$$\mathrm{tr}\left(\varphi \left(A\right)\tilde{\psi }\left(B\right)\right)=\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B^{1/k}\right)}^k\right)=\mathrm{tr}\left(\mathit{AB}\right),A\in {\mathcal{M}}_n,B\in S^{\prime }.$$E44

The complex span of $S^{\prime }$ is ${\mathcal{M}}_n$. By Theorem 2.2, $\varphi$ is bijective linear, and $\tilde{\psi }$ can be extended to a linear bijection on ${\mathcal{M}}_n$.

The linearity of $\psi$ and (40) imply that for every $B\in {\mathcal{M}}_n$, there exists $\mathrm{ϵ}>0$ such that $I_n+\mathit{xB}\in S$ and the power series of ${\left(I_n+\mathit{xB}\right)}^k$ converges whenever $\mid x\mid <\mathrm{ϵ}$. Then

$$\mathrm{tr}\left(\varphi \left(A\right){\left(\psi \left(I_n\right)+\mathit{x\psi }\left(B\right)\right)}^k\right)=\mathrm{tr}\left(A{\left(I_n+\mathit{xB}\right)}^k\right),A\in {\mathcal{M}}_n,\mid x\mid <\mathrm{ϵ}.$$E45

1. 1. First suppose $k\ge 2$. Equating degree one terms and degree $\left(k-1\right)$ terms on both sides of (45) respectively, we get the following identities for $A,B\in {\mathcal{M}}_n$:

$$\mathrm{tr}\left(\varphi \left(A\right)\left(\sum _{i=0}^{k-1}\psi {\left(I_n\right)}^{k-1-i}\psi \left(B\right)\psi {\left(I_n\right)}^i\right)\right)=\mathrm{tr}\left(\mathit{kAB}\right),$$E46

$$\mathrm{tr}\left(\varphi \left(A\right)\left(\sum _{i=0}^{k-1}\psi {\left(B\right)}^i\psi \left(I_n\right)\psi {\left(B\right)}^{k-1-i}\right)\right)=\mathrm{tr}\left({\mathit{kAB}}^{k-1}\right).$$E47

Let $\left\{C_i:i=1,\dots ,n^2\right\}$ be a basis of projection matrices (i.e. $C_i^2=C_i$) in ${\mathcal{M}}_n$. For example, we may choose the following basis of rank 1 projections:

$$\left\{E_{\mathit{ii}}:1\le i\le n\right\}\cup \left\{\frac{1}{\sqrt{2}}\left(E_{\mathit{ii}}+E_{\mathit{jj}}+\delta E_{\mathit{ij}}+\overline{\delta }{E}_{\mathit{ji}}\right):1\le i<j\le n,\delta \in \{1,\mathbf{i}\}\right\}.$$E48

By (40) and (47), for $A\in {\mathcal{M}}_n$ and $i=1,\dots ,n^2$,

$$\mathrm{tr}\left(\mathit{k\varphi }\left(A\right)\psi {\left(C_i\right)}^k\right)=\mathrm{tr}\left({\mathit{kAC}}_i\right)=\mathrm{tr}\left(\varphi \left(A\right)\left(\sum _{j=0}^{k-1}\psi {\left(C_i\right)}^j\psi \left(I_n\right)\psi {\left(C_i\right)}^{k-1-j}\right)\right).$$E49

By the bijectivity of $\varphi$,

$$\mathit{k\psi }{\left(C_i\right)}^k=\sum _{j=0}^{k-1}\psi {\left(C_i\right)}^j\psi \left(I\right)\psi {\left(C_i\right)}^{k-1-j}.$$E50

Therefore, for $i=1,\dots ,n^2$,

$$\begin{array}{c}0=\psi \left(C_i\right)\left(\sum _{j=0}^{k-1}\psi {\left(C_i\right)}^j\psi \left(I_n\right)\psi {\left(C_i\right)}^{k-1-j}\right)-\left(\sum _{j=0}^{k-1}\psi {\left(C_i\right)}^j\psi \left(I_n\right)\psi {\left(C_i\right)}^{k-1-j}\right)\psi \left(C_i\right)\\ =\psi {\left(C_i\right)}^k\psi \left(I_n\right)-\psi \left(I_n\right)\psi {\left(C_i\right)}^k.\end{array}$$E51

Since

$$\mathrm{tr}\left(\mathit{A\psi }{\left(C_i\right)}^k\right)=\mathrm{tr}\left({\varphi }^{-1}\left(A\right)C_i^k\right)=\mathrm{tr}\left({\varphi }^{-1}\left(A\right)C_i\right),A\in {\mathcal{M}}_n,i=1,\dots ,n^2,$$E52

the only matrix $A\in {\mathcal{M}}_n$ such that $\mathrm{tr}\left(\mathit{A\psi }{\left(C_i\right)}^k\right)=0$ for all $i\in \left\{1,\dots ,n^2\right\}$ is the zero matrix. So $\left\{\psi {\left(C_i\right)}^k:i=1,\dots ,n^2\right\}$ is a basis of ${\mathcal{M}}_n$. (51) implies that $\psi \left(I_n\right)={\mathit{cI}}_n$ for certain $c\in \mathbb{C}\backslash \left\{0\right\}$.

(46) shows that

$$c^{k-1}\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right),A,B\in {\mathcal{M}}_n.$$E53

Therefore,

$$c^{k-1}\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B^k\right)\right)=\mathrm{tr}\left({\mathit{AB}}^k\right)=\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right),A\in {\mathcal{M}}_n,B\in S.$$E54

The bijectivity of $\varphi$ shows that $c^{k-1}\psi \left(B^k\right)=\psi {\left(B\right)}^k$ for $B\in S$, that is,

$$c^{-1}\psi \left(B^k\right)={\left[c^{-1}\psi \left(B\right)\right]}^k,B\in S.$$E55

Notice that $c^{-1}\psi \left(I_n\right)=I_n$. By Theorem 3.4, there is an invertible $P\in {\mathcal{M}}_n$ such that $\psi$ is of the form $\psi \left(B\right)={\mathit{\text{cPBP}}}^{-1}$ or $\psi \left(B\right)={\mathit{cPB}}^t{P}^{-1}$ for $B\in {\mathcal{M}}_n$. Consequently, we get (42).

1. 2. Now suppose $k<0$. Then $\psi \left(I_n\right)$ is invertible. For every $B\in {\mathcal{M}}_n$ and sufficiently small $x$, we have the power series expansion:

$$\begin{array}{l}{\left(\psi \left(I_n\right)+\mathit{x\psi }\left(B\right)\right)}^k\\ ={\left[{\left(I_n+\mathit{x\psi }{\left(I_n\right)}^{-1}\psi \left(B\right)\right)}^{-1}\psi {\left(I_n\right)}^{-1}\right]}^{\mid k\mid }\\ ={\left[\left(I_n-\mathit{x\psi }{\left(I_n\right)}^{-1}\psi \left(B\right)+x^2\psi {\left(I_n\right)}^{-1}\psi \left(B\right)\psi {\left(I_n\right)}^{-1}\psi \left(B\right)+\cdots \right)\psi {\left(I_n\right)}^{-1}\right]}^{\mid k\mid }\\ =\psi {\left(I_n\right)}^k-x\left(\sum _{i=1}^{\mid k\mid }\psi {\left(I_n\right)}^{-i}\psi \left(B\right)\psi {\left(I_n\right)}^{k-1+i}\right)\\ +x^2\left(\sum _{i=1}^{\mid k\mid }\sum _{j=1}^{\mid k\mid +1-i}\psi {\left(I_n\right)}^{-i}\psi \left(B\right)\psi {\left(I_n\right)}^{-j}\psi \left(B\right)\psi {\left(I_n\right)}^{k-2+i+j}\right)+\cdots \end{array}$$E56

Equating degree one terms and degree two terms of (45) respectively and using (56), we get the following identities for $A,B\in {\mathcal{M}}_n$:

$$\mathrm{tr}\left(\varphi \left(A\right)\left(\sum _{i=1}^{\mid k\mid }\psi {\left(I_n\right)}^{-i}\psi \left(B\right)\psi {\left(I_n\right)}^{k-1+i}\right)\right)=\mathrm{tr}\left(|k|\mathit{AB}\right),$$E57

$$\mathrm{tr}\left(\varphi \left(A\right)\left(\sum _{i=1}^{\mid k\mid }\sum _{j=1}^{\mid k\mid +1-i}\psi {\left(I_n\right)}^{-i}\psi \left(B\right)\psi {\left(I_n\right)}^{-j}\psi \left(B\right)\psi {\left(I_n\right)}^{k-2+i+j}\right)\right)=\mathrm{tr}\left(\frac{k\left(k-1\right)}{2}{\mathit{AB}}^2\right).$$E58

(57) and (40) imply that

$$\sum _{i=1}^{\mid k\mid }\psi {\left(I_n\right)}^{-i}\psi \left(B^k\right)\psi {\left(I_n\right)}^{k-1+i}=\mid k\mid \psi {\left(B\right)}^k,B\in S.$$E59

Let $F_r\left(B\right)$ denote the degree $r$ coefficient in the power series of ${\left(\psi \left(I_n\right)+\mathit{x\psi }\left(B\right)\right)}^k$. Then (57) and (58) show that:

$$\frac{k-1}{2}{F}_1\left(B^2\right)=F_2\left(B\right),B\in {\mathcal{M}}_n.$$E60

Denote ${\psi }_1\left(B\right)≔\psi \left(B\right)\psi {\left(I_n\right)}^{-1}$. We discuss the cases $k=-1$ and $k\ne -1$.

1. When $k=-1$, (60) leads to

$$\psi \left(B^2\right)=\psi \left(B\right)\psi {\left(I_n\right)}^{-1}\psi \left(B\right),B\in {\mathcal{M}}_n.$$E61

So ${\psi }_1\left(B^2\right)={\psi }_1{\left(B\right)}^2$ for $B\in {\mathcal{M}}_n$. Note that ${\psi }_1\left(I_n\right)=I_n$. By Theorem 3.4, there exists an invertible $P\in {\mathcal{M}}_n$ such that ${\psi }_1\left(B\right)={\mathit{PBP}}^{-1}$ or ${\psi }_1\left(B\right)={\mathit{PB}}^t{P}^{-1}$ for $B\in {\mathcal{M}}_n$. Let $Q≔P^{-1}\psi \left(I_n\right)$. Then $Q$ is invertible, and $\psi \left(B\right)=\mathit{PBQ}$ or $\psi \left(B\right)={\mathit{PB}}^{t}Q$ for $B\in {\mathcal{M}}_n$. Using (40), we get (41).

1. b. Suppose the integer $k<-1$. Then (60) implies that

$$\frac{k-1}{2}\left(\psi {\left(I_n\right)}^{-1}{F}_1\left(B^2\right)-F_2\left(B^2\right)\psi {\left(I_n\right)}^{-1}\right)=\psi {\left(I_n\right)}^{-1}{F}_2\left(B\right)-F_2\left(B\right)\psi {\left(I_n\right)}^{-1},$$E62

which gives

$$\begin{array}{l}\frac{1-k}{2}\left(\psi {\left(I_n\right)}^k\psi \left(B^2\right)-\psi \left(B^2\right)\psi {\left(I_n\right)}^k\right)\\ =\psi {\left(I_n\right)}^k\psi \left(B\right)\psi {\left(I_n\right)}^{-1}\psi \left(B\right)-\psi \left(B\right)\psi {\left(I_n\right)}^{-1}\psi \left(B\right)\psi {\left(I_n\right)}^k.\end{array}$$E63

In other words, for $B\in {\mathcal{M}}_n$:

$$\psi {\left(I_n\right)}^k\left(\frac{1-k}{2}{\psi }_1\left(B^2\right)-{\psi }_1{\left(B\right)}^2\right)=\left(\frac{1-k}{2}{\psi }_1\left(B^2\right)-{\psi }_1{\left(B\right)}^2\right)\psi {\left(I_n\right)}^k.$$E64

Let $B=I_n+\mathit{xE}$ for an arbitrary matrix $E\in {\mathcal{M}}_n$. Then (64) becomes

$$\begin{array}{l}\psi {\left(I_n\right)}^k\left[x\left(-1-k\right){\psi }_1\left(E\right)+x^2\left(\frac{1-k}{2}{\psi }_1\left(E^2\right)-{\psi }_1{\left(E\right)}^2\right)\right]\\ =\left[x\left(-1-k\right){\psi }_1\left(E\right)+x^2\left(\frac{1-k}{2}{\psi }_1\left(E^2\right)-{\psi }_1{\left(E\right)}^2\right)\right]\psi {\left(I_n\right)}^k.\end{array}$$E65

The equality on degree one terms shows that $\psi {\left(I_n\right)}^k$ commutes with all ${\psi }_1\left(E\right)$. Hence $\psi {\left(I_n\right)}^k$ commutes with the range of $\psi$. (??) can be rewritten as

$$\mathrm{tr}\left(\left(\sum _{i=1}^{\mid k\mid }\psi {\left(I_n\right)}^{k-1+i}\varphi \left(A\right)\psi {\left(I_n\right)}^{-i}\right)\psi \left(B\right)\right)=\mathrm{tr}\left(|k|\mathit{AB}\right),A,B\in {\mathcal{M}}_n.$$E66

By Theorem 2.2, $\psi$ is a linear bijection and its range is ${\mathcal{M}}_n$. So $\psi {\left(I_n\right)}^k=\mu I_n$ for certain $\mu \in \mathbb{C}$.

Now by (59), for $B\in S$:

$$\begin{array}{l}\mid k\mid \psi \left(I_n\right)\psi {\left(B\right)}^k-\mid k\mid \psi {\left(B\right)}^k\psi \left(I_n\right)\\ =\psi \left(I_n\right)\left(\sum _{i=1}^{\mid k\mid }\psi {\left(I_n\right)}^{-i}\psi \left(B^k\right)\psi {\left(I_n\right)}^{k-1+i}\right)-\left(\sum _{i=1}^{\mid k\mid }\psi {\left(I_n\right)}^{-i}\psi \left(B^k\right)\psi {\left(I_n\right)}^{k-1+i}\right)\psi \left(I_n\right)\\ =\psi \left(B^k\right)\psi {\left(I_n\right)}^k-\psi {\left(I_n\right)}^k\psi \left(B^k\right)=0\end{array}$$E67

So $\psi \left(I_n\right)$ commutes with $\psi {\left(B\right)}^k$ for $B\in S$. In particular, $\psi \left(I_n\right)$ commutes with $\tilde{\psi }\left(B\right)=\psi {\left(B^{1/k}\right)}^k$ for $B\in S^{\prime }$. The complex span of $S^{\prime }$ is ${\mathcal{M}}_n$, and $\tilde{\psi }$ can be extended to a linear bijection on ${\mathcal{M}}_n$. Hence $\psi \left(I_n\right)={\mathit{cI}}_n$ for certain $c\in \mathbb{C}\backslash \left\{0\right\}$. By (59), we get ${\psi }_1\left(B^k\right)={\psi }_1{\left(B\right)}^k$ for $B\in S$. Note that ${\psi }_1\left(I_n\right)=I_n$. By Theorem 3.4, there is an invertible $P\in {\mathcal{M}}_n$ such that ${\psi }_1\left(B\right)={\mathit{PBP}}^{-1}$ or ${\psi }_1\left(B\right)={\mathit{PB}}^t{P}^{-1}$. Then $\psi \left(B\right)={\mathit{\text{cPBP}}}^{-1}$ or $\psi \left(B\right)={\mathit{cPB}}^t{P}^{-1}$. Using (40), we get (42).

Remark 3.6 The following modifications could be applied to the proof of Theorem 3.5 for $\mathbb{F}=\mathbb{R}$:

1. Let $S^{\prime }$ be the collection of matrices $A={\mathit{QDQ}}^{-1}$, in which $D$ is nonnegative diagonal and $Q\in {\mathcal{M}}_n\left(\mathbb{R}\right)$ is invertible, such that $A^{1/k}={\mathit{QD}}^{1/k}{Q}^{-1}\in S$.

2. We may choose the following basis of rank 1 projections of ${\mathcal{M}}_n\left(\mathbb{R}\right)$ to substitute (48):

$$\begin{array}{l}\left\{E_{\mathit{ii}}:1\le i\le n\right\}\cup \left\{\frac{1}{\sqrt{2}}\left(E_{\mathit{ii}}+E_{\mathit{jj}}+E_{\mathit{ij}}+E_{\mathit{ji}}\right):1\le i<j\le n\right\}\\ \cup \left\{{\omega }_1{E}_{\mathit{ii}}+{\omega }_2{E}_{\mathit{jj}}+E_{\mathit{ij}}-E_{\mathit{ji}}:1\le i<j\le n\right\},\end{array}$$E68

in which ${\omega }_1,{\omega }_2$ are distinct roots of $x^2-x-1=0.$.

The arguments in the above proof will be applied analogously to maps on the other sets discussed in this paper.

4. $k$-power linear preservers and trace of power-product preservers on ${\mathcal{H}}_n$

4.1 $k$-power linear preservers on ${\mathcal{H}}_n$

We give a result that determine linear operators on ${\mathcal{H}}_n$ that satisfy $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ on a neighborhood of $I_n$ in ${\mathcal{H}}_n$ for certain $k\in \mathbb{Z}\backslash \left\{0,1\right\}$

Theorem 4.1. Fix $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. A nonzero linear map $\psi :{\mathcal{H}}_n\to {\mathcal{H}}_n$ satisfies that

$$\psi \left(A^k\right)=\psi {\left(A\right)}^k$$E69

on an open neighborhood of $I_n$ consisting of invertible matrices if and only if $\psi$ is of the following forms for certain unitary matrix $U\in {\mathcal{M}}_n$:

1. When $k$ is even,

$$\psi \left(A\right)=U^{\ast }\mathit{AU},A\in {\mathcal{H}}_n;\mathit{\text{or}}\psi \left(A\right)=U^{\ast }{A}^{t}U,A\in {\mathcal{H}}_n.$$E70

1. 2. When $k$ is odd,

$$\psi \left(A\right)=\pm U^{\ast }\mathit{AU},A\in {\mathcal{H}}_n;\mathit{\text{or}}\psi \left(A\right)=\pm U^{\ast }{A}^{t}U,A\in {\mathcal{H}}_n.$$E71

Proof. It suffices to prove the necessary part. Suppose (69) holds on an open neighborhood $S$ of $I_n$ in ${\mathcal{H}}_n$.

1. 1. First assume $k\ge 2$. Replacing ${\mathcal{M}}_n$ by ${\mathcal{H}}_n$ in part (1) of the proof of Theorem 3.4 up to (33), we can prove that $\psi \left(I_n\right)$ commutes with the range of $\psi$, and ${\psi }_1\left(A\right)≔\psi {\left(I_n\right)}^{k-2}\psi \left(A\right)$ is a nonzero linear map that satisfies ${\psi }_1\left(A^2\right)={\psi }_1{\left(A\right)}^2$ for $A\in {\mathcal{H}}_n$.

Every matrix in ${\mathcal{M}}_n$ can be uniquely expressed as $A+\mathbf{i}B$ for $A,B\in {\mathcal{H}}_n$. Extend ${\psi }_1$ to a map $\tilde{\psi }:{\mathcal{M}}_n\to {\mathcal{M}}_n$ such that

$$\tilde{\psi }\left(A+\mathbf{i}B\right)={\psi }_1\left(A\right)+\mathbf{i}{\psi }_1\left(B\right),A,B\in {\mathcal{H}}_n.$$E72

It is straightforward to check that $\tilde{\psi }$ is a complex linear bijection. Moreover, for $A,B\in {\mathcal{H}}_n$,

$$\begin{array}{c}{\psi }_1\left(\mathit{AB}+\mathit{BA}\right)={\psi }_1\left({\left(A+B\right)}^2\right)-{\psi }_1\left(A^2\right)-{\psi }_1\left(B^2\right)\\ ={\psi }_1{\left(A+B\right)}^2-{\psi }_1{\left(A\right)}^2-{\psi }_1{\left(B\right)}^2\\ ={\psi }_1\left(A\right){\psi }_1\left(B\right)+{\psi }_1\left(B\right){\psi }_1\left(A\right).\end{array}$$

It implies that

$$\tilde{\psi }\left({\left(A+\mathbf{i}B\right)}^2\right)=\tilde{\psi }{\left(A+\mathbf{i}B\right)}^2,A,B\in {\mathcal{H}}_n.$$

By Theorem 3.1, there is an invertible matrix $U\in {\mathcal{M}}_n$ such that

1. $\tilde{\psi }\left(A\right)={\mathit{UAU}}^{-1}$ for all $A\in {\mathcal{M}}_n$, or

2. $\tilde{\psi }\left(A\right)={\mathit{UA}}^t{U}^{-1}$ for all $A\in {\mathcal{M}}_n$.

First suppose $\tilde{\psi }\left(A\right)={\mathit{UAU}}^{-1}$. The restriction of $\tilde{\psi }$ on ${\mathcal{H}}_n$ is ${\psi }_1:{\mathcal{H}}_n\to {\mathcal{H}}_n$. Hence for $A\in {\mathcal{H}}_n$, we have ${\mathit{UAU}}^{-1}={\left({\mathit{UAU}}^{-1}\right)}^{\ast }=U^{-\ast }{\mathit{AU}}^{\ast }$; then $U^{\ast }\mathit{UA}={\mathit{AU}}^{\ast }U$ for all $A\in {\mathcal{H}}_n$, which shows that $U^{\ast }U={\mathit{cI}}_n$ for certain $c\in {\mathbb{R}}^{+}.$ By adjusting a scalar if necessary, we may assume that $U$ is unitary. So $\psi {\left(I_n\right)}^{k-2}\psi \left(A\right)={\mathit{UAU}}^{\ast }$. Then $\psi {\left(I_n\right)}^{k-1}=I_n$, so that $\psi \left(I_n\right)=I_n$ when $k$ is even and $\psi \left(I_n\right)\in \left\{I_n,-I_n\right\}$ when $k$ is odd. Thus $\psi \left(A\right)={\mathit{UAU}}^{\ast }$ when $k$ is even and $\psi \left(A\right)=\pm {\mathit{UAU}}^{\ast }$ when $k$ is odd. Similarly for the case $\tilde{\psi }\left(A\right)={\mathit{UA}}^t{U}^{-1}$. Therefore, (70) or (71) holds.

1. 2. Now assume that $k<0$. Replacing ${\mathcal{M}}_n$ by ${\mathcal{H}}_n$ in part (2) of the proof of Theorem 3.4, we can show that $\psi \left(I_n\right)$ commutes with the range of $\psi$, and furthermore the nonzero linear map ${\psi }_1\left(A\right)≔\psi {\left(I_n\right)}^{-1}\psi \left(A\right)$ satisfies that ${\psi }_1\left(A^2\right)={\psi }_1{\left(A\right)}^2$. By arguments in the preceding paragraphs, we get (70) or (71).

4.2 Trace of power-product preservers on ${\mathcal{H}}_n$

By Corollary 2.3, every linear bijection $\varphi :{\mathcal{H}}_n\to {\mathcal{H}}_n$ corresponds to another linear bijection $\psi :{\mathcal{H}}_n\to {\mathcal{H}}_n$ such that $\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right)$ for all $A,B\in {\mathcal{H}}_n$. When $m\ge 3$, linear maps ${\varphi }_1,\cdots ,{\varphi }_m:{\mathcal{H}}_n\to {\mathcal{H}}_n$ that satisfy $\mathrm{tr}\left({\varphi }_1\left(A_1\right)\cdots {\varphi }_m\left(A_m\right)\right)=\mathrm{tr}\left(A_1\cdots A_m\right)$ are characterized in [18].

Theorem 4.2. Let $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. Let $S$ be an open neighborhood of $I_n$ in ${\mathcal{H}}_n$ consisting of invertible Hermitian matrices. Then two maps $\varphi ,\psi :{\mathcal{H}}_n\to {\mathcal{H}}_n$ satisfy that

$$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left({\mathit{AB}}^k\right),$$E73

1. for all $A\in {\mathcal{H}}_n,B\in S$, and $\psi$ is linear, or

2. for all $A,B\in S$ and both $\varphi$ and $\psi$ are linear,

if and only if $\varphi$ and $\psi$ take the following forms:

1. When $k=-1$, there exist an invertible matrix $P\in {\mathcal{M}}_n$ and $c\in \left\{1,-1\right\}$ such that

$$\left(\begin{array}{c}\varphi \left(A\right)={\mathit{cP}}^{\ast }\mathit{AP}\\ \psi \left(B\right)={\mathit{cP}}^{\ast }\mathit{BP}\end{array}\right.A,B\in {\mathcal{H}}_n;\mathit{\text{or}}\left(\begin{array}{c}\varphi \left(A\right)={\mathit{cP}}^{\ast }{A}^{t}P\\ \psi \left(B\right)={\mathit{cP}}^{\ast }{B}^{t}P\end{array}\right.A,B\in {\mathcal{H}}_n.$$E74

1. b. When $k\in \mathbb{Z}\backslash \left\{-\mathrm{1,0,1}\right\}$, there exist a unitary matrix $U\in {\mathcal{M}}_n$ and $c\in \mathbb{R}\backslash \left\{0\right\}$ such that

$$\left(\begin{array}{c}\varphi \left(A\right)=c^{-k}{U}^{\ast }\mathit{AU}\\ \psi \left(B\right)={\mathit{cU}}^{\ast }\mathit{BU}\end{array}\right.A,B\in {\mathcal{H}}_n;\mathit{\text{or}}\left(\begin{array}{c}\varphi \left(A\right)=c^{-k}{U}^{\ast }{A}^{t}U\\ \psi \left(B\right)={\mathit{cU}}^{\ast }{B}^{t}U\end{array}\right.A,B\in {\mathcal{H}}_n.$$E75

Proof. Assumption (2) leads to assumption (1) (cf. the proof of Theorem 3.5). We prove the theorem under assumption (1). It suffices to prove the necessary part.

1. When $k\ge 2$, in the part (1) of proof of Theorem 3.5, through replacing ${\mathcal{M}}_n$ by ${\mathcal{H}}_n$, complex numbers by real numbers, and Theorem 3.1 or Theorem 3.4 by Theorem 4.1, we can prove that $\left(\varphi ,\psi \right)$ has the forms in (75).

2. When $k<0$, in the part (2) of proof of Theorem 3.5, through replacing ${\mathcal{M}}_n$ by ${\mathcal{H}}_n$ and complex numbers by real numbers, we can get the corresponding equalities of (56) $\sim$ (60) on ${\mathcal{H}}_n$. The case $k<-1$ can be proved completely analogously with the help of Theorem 4.1.

For the case $k=-1$, the equality corresponding to (60) can be simplified as

$$\psi \left(B^2\right)=\psi \left(B\right)\psi {\left(I_n\right)}^{-1}\psi \left(B\right),B\in {\mathcal{H}}_n.$$E76

Let ${\psi }_1\left(B\right)≔\psi {\left(I_n\right)}^{-1}\psi \left(B\right)$. Then ${\psi }_1:{\mathcal{H}}_n\to {\mathcal{M}}_n$ is a nonzero real linear map that satisfies ${\psi }_1\left(B^2\right)={\psi }_1{\left(B\right)}^2$ for $B\in {\mathcal{H}}_n$. Extend ${\psi }_1$ to a complex linear map $\tilde{\psi }:{\mathcal{M}}_n\to {\mathcal{M}}_n$ such that

$$\tilde{\psi }\left(A+\mathbf{i}B\right)≔{\psi }_1\left(A\right)+\mathbf{i}{\psi }_1\left(B\right),A,B\in {\mathcal{H}}_n.$$E77

Similarly to the arguments in part (1) of the proof of Theorem 4.1, we have $\tilde{\psi }\left({\left(A+\mathbf{i}B\right)}^2\right)={\left(\tilde{\psi }\left(A+\mathbf{i}B\right)\right)}^2$ for all $A,B\in {\mathcal{H}}_n$. Using Theorem 3.4 and the fact that $\tilde{\psi }\left(I_n\right)={\psi }_1\left(I_n\right)=I_n$, we can prove that there is an invertible $P\in {\mathcal{M}}_n$ such that for all $B\in {\mathcal{H}}_n$, either ${\psi }_1\left(B\right)=P^{-1}\mathit{BP}$ or ${\psi }_1\left(B\right)=P^{-1}{B}^{t}P$. So

$$\psi \left(B\right)=\psi \left(I_n\right)P^{-1}\mathit{BP},B\in {\mathcal{H}}_n;\text{or}$$E78

$$\psi \left(B\right)=\psi \left(I_n\right)P^{-1}{B}^{t}P,B\in {\mathcal{H}}_n.$$E79

If $\psi \left(B\right)=\psi \left(I_n\right)P^{-1}\mathit{BP}$ for $B\in {\mathcal{H}}_n$, then $\psi \left(I_n\right)P^{-1}\mathit{BP}={\left(\psi \left(I_n\right)P^{-1}\mathit{BP}\right)}^{\ast }=P^{\ast }{\mathit{BP}}^{-\ast }\psi \left(I_n\right),$ which gives

$$\left(P^{-\ast }\psi \left(I_n\right)P^{-1}\right)B=B\left(P^{-\ast }\psi \left(I_n\right)P^{-1}\right),B\in {\mathcal{H}}_n.$$E80

Hence $P^{-\ast }\psi \left(I_n\right)P^{-1}={\mathit{cI}}_n$ for certain $c\in \mathbb{R}\backslash \left\{0\right\}.$ We have $\psi \left(I_n\right)={\mathit{cP}}^{\ast }P$ so that $\psi \left(B\right)={\mathit{cP}}^{\ast }\mathit{BP}$ for $B\in {\mathcal{H}}_n$. Similarly for the case $\psi \left(B\right)=\psi \left(I_n\right)P^{-1}{B}^{t}P$. Adjusting $c$ and $P$ by scalar factors simultaneously, we may assume that $c\in \left\{1,-1\right\}$. It implies (74).

Remark 4.3. Theorem 4.2 does not hold if $\psi$ is not assumed to be linear. Let $k$ be a positive even integer. Let $\tilde{\psi }:{\mathcal{H}}_n\to {\mathcal{H}}_n$ be any bijective linear map such that $\tilde{\psi }\left(\overline{{\mathcal{P}}_n}\right)\subseteq \overline{{\mathcal{P}}_n}$. For example, $\tilde{\psi }$ may be a completely positive map of the form $\tilde{\psi }\left(B\right)={\sum }_{i=1}^r{N}_i^{\ast }{\mathit{BN}}_i$ for $r\ge 2$, $N_1,\dots ,N_r\in {\mathcal{M}}_n$ linearly independent, and at least one of $N_1,\dots ,N_r$ is invertible. By Corollary 2.3, there is a linear bijection $\varphi :{\mathcal{H}}_n\to {\mathcal{H}}_n$ such that $\mathrm{tr}\left(\varphi \left(A\right)\tilde{\psi }\left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right)$ for all $A,B\in {\mathcal{H}}_n$. Let $\psi :{\mathcal{H}}_n\to {\mathcal{H}}_n$ be defined by $\psi \left(B\right)=\tilde{\psi }{\left(B^k\right)}^{1/k}.$ Then

$$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left(\varphi \left(A\right)\tilde{\psi }\left(B^k\right)\right)=\mathrm{tr}\left({\mathit{AB}}^k\right),A,B\in {\mathcal{H}}_n.$$

Obviously, $\psi$ may be non-linear, and the choices of pairs $\left(\varphi ,\psi \right)$ are much more than those in (74) and (75).

5. $k$-power linear preservers and trace of power-product preservers on ${\mathcal{S}}_n$ and ${\mathcal{S}}_n\left(\mathbb{R}\right)$

5.1 $k$-power linear preservers on ${\mathcal{S}}_n$ and ${\mathcal{S}}_n\left(\mathbb{R}\right)$

Chan and Lim described the linear $k$-power preservers on ${\mathcal{S}}_n\left(\mathbb{F}\right)$ for $k\ge 2$ in [7, Theorem 2] as follows.

Theorem 5.1. (Chan, Lim [5]) Let an integer $k\ge 2$. Let $\mathbb{F}$ be an algebraic closed field with $\mathit{char}\left(\mathbb{F}\right)=0$ or $\mathit{char}\left(\mathbb{F}\right)>k$. Suppose that $\psi :{\mathcal{S}}_n\left(\mathbb{F}\right)\to {\mathcal{S}}_n\left(\mathbb{F}\right)$ is a nonzero linear operator such that $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A\in {\mathcal{S}}_n\left(\mathbb{F}\right)$. Then there exist $\lambda \in \mathbb{F}$ with ${\lambda }^{k-1}=1$ and an orthogonal matrix $O\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\psi \left(A\right)=\lambda {\mathit{OAO}}^t,A\in {\mathcal{S}}_n.$$E81

We generalize Theorem 5.1 to include the case ${\mathcal{S}}_n\left(\mathbb{R}\right)$, to include negative integers $k$, and to assume the $k$-power preserving condition only on matrices nearby the identity.

Theorem 5.2. Let $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Suppose that $\psi :{\mathcal{S}}_n\left(\mathbb{F}\right)\to {\mathcal{S}}_n\left(\mathbb{F}\right)$ is a nonzero linear map such that $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A$ in an open neighborhood of $I_n$ in ${\mathcal{S}}_n\left(\mathbb{F}\right)$ consisting of invertible matrices. Then there exist $\lambda \in \mathbb{F}$ with ${\lambda }^{k-1}=1$ and an orthogonal matrix $O\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\psi \left(A\right)=\lambda {\mathit{OAO}}^t,A\in {\mathcal{S}}_n\left(\mathbb{F}\right).$$E82

Proof. It suffices to prove the necessary part. In both $k\ge 2$ and $k<0$ cases, using analogous arguments as parts (1) and (2) of the proof of Theorem 3.4, we get that $\psi \left(I_n\right)$ commutes with the range of $\psi$, and the nonzero map ${\psi }_1\left(A\right)≔\psi {\left(I_n\right)}^{k-2}\psi \left(A\right)$ satisfies that ${\psi }_1\left(A^2\right)={\psi }_1{\left(A\right)}^2$ for $A\in {\mathcal{S}}_n\left(\mathbb{F}\right)$. Then

$$\begin{array}{c}{\psi }_1\left(A\right){\psi }_1\left(B\right)+{\psi }_1\left(B\right){\psi }_1\left(A\right)={\psi }_1{\left(A+B\right)}^2-{\psi }_1{\left(A\right)}^2-{\psi }_1{\left(B\right)}^2\\ ={\psi }_1\left({\left(A+B\right)}^2\right)-{\psi }_1\left(A^2\right)-{\psi }_1\left(B^2\right)\\ ={\psi }_1\left(\mathit{AB}+\mathit{BA}\right).\end{array}$$E83

In particular, ${\psi }_1\left(A\right){\psi }_1\left(A^r\right)+{\psi }_1\left(A^r\right){\psi }_1\left(A\right)=2{\psi }_1\left(A^{r+1}\right)$ for $r\in {\mathbb{Z}}_{+}$. Using induction, we get ${\psi }_1\left(A^{\mathrm{\ell }}\right)={\psi }_1{\left(A\right)}^{\mathrm{\ell }}$ for all $A\in {\mathcal{S}}_n\left(\mathbb{F}\right)$ and $\mathrm{\ell }\in {\mathbb{Z}}_{+}$. By [26, Corollary 6.5.4], there is an orthogonal matrix $O\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that ${\psi }_1\left(A\right)={\mathit{OAO}}^t$. Since $\psi \left(I_n\right)$ commutes with the range of ${\psi }_1$, we have $\psi \left(I_n\right)=\lambda I_n$ for certain $\lambda \in \mathbb{F}$ in which ${\lambda }^{k-1}=1$. So $\psi \left(A\right)=\lambda {\mathit{OAO}}^t$ as in (82).

Obviously, in $\mathbb{F}=\mathbb{R}$ case, (82) has $\lambda =1$ when $k$ is even and $\lambda \in \left\{1,-1\right\}$ when $k$ is odd.

5.2 Trace of power-product preservers on ${\mathcal{S}}_n$ and ${\mathcal{S}}_n\left(\mathbb{R}\right)$

Corollary 2.3 shows that every linear bijection $\varphi :{\mathcal{S}}_n\left(\mathbb{F}\right)\to {\mathcal{S}}_n\left(\mathbb{F}\right)$ corresponds to another linear bijection $\psi :{\mathcal{S}}_n\left(\mathbb{F}\right)\to {\mathcal{S}}_n\left(\mathbb{F}\right)$ such that $\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right)$ for all $A,B\in {\mathcal{S}}_n\left(\mathbb{F}\right)$. When $m\ge 3$, maps ${\varphi }_1,\cdots ,{\varphi }_m:{\mathcal{S}}_n\left(\mathbb{F}\right)\to {\mathcal{S}}_n\left(\mathbb{F}\right)$ that satisfy $\mathrm{tr}\left({\varphi }_1\left(A_1\right)\cdots {\varphi }_m\left(A_m\right)\right)=\mathrm{tr}\left(A_1\cdots A_m\right)$ are determined in [18].

We characterize the trace of power-product preserver for ${\mathcal{S}}_n\left(\mathbb{F}\right)$ here.

Theorem 5.3. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. Let $S$ be an open neighborhood of $I_n$ in ${\mathcal{S}}_n\left(\mathbb{F}\right)$ consisting of invertible matrices. Then two maps $\varphi ,\psi :{\mathcal{S}}_n\left(\mathbb{F}\right)\to {\mathcal{S}}_n\left(\mathbb{F}\right)$ satisfy that

$$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left({\mathit{AB}}^k\right),$$E84

1. for all $A\in {\mathcal{S}}_n\left(\mathbb{F}\right),B\in S$, and $\psi$ is linear, or

2. for all $A,B\in S$ and both $\varphi$ and $\psi$ are linear,

if and only if $\varphi$ and $\psi$ take the following forms:

1. When $k=-1$, there exist an invertible matrix $P\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ and $c\in \mathbb{F}\backslash \left\{0\right\}$ such that

$$\varphi \left(A\right)={\mathit{\text{cPAP}}}^t,\psi \left(B\right)={\mathit{\text{cPBP}}}^t,A,B\in {\mathcal{S}}_n\left(\mathbb{F}\right).$$E85

We may choose $c=1$ for $\mathbb{F}=\mathbb{C}$ and $c\in \left\{1,-1\right\}$ for $\mathbb{F}=\mathbb{R}$.

1. b. When $k\in \mathbb{Z}\backslash \left\{-\mathrm{1,0,1}\right\}$, there exist $c\in \mathbb{F}\backslash \left\{0\right\}$ and an orthogonal matrix $O\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\varphi \left(A\right)=c^{-k}{\mathit{OAO}}^t,\psi \left(B\right)={\mathit{\text{cOBO}}}^t,A,B\in {\mathcal{S}}_n\left(\mathbb{F}\right).$$E86

Proof. Assumption (2) leads to assumption (1) (cf. the proof of Theorem 3.5). We prove the theorem under assumption (1). It suffices to prove the necessary part.

Obviously, ${\mathcal{S}}_n\cap {\mathcal{H}}_n={\mathcal{S}}_n\left(\mathbb{R}\right)$ and ${\mathcal{S}}_n\cap {\mathcal{P}}_n={\mathcal{P}}_n\left(\mathbb{R}\right)$. Let $S^{\prime }≔\left\{B\in {\mathcal{P}}_n\left(\mathbb{R}\right):B^{1/k}\in S\right\}$, which is an open neighborhood of $I_n$ in ${\mathcal{P}}_n\left(\mathbb{R}\right)$ and whose real (resp. complex) span is ${\mathcal{S}}_n\left(\mathbb{R}\right)$ (resp. ${\mathcal{S}}_n$). Using an analogous argument of the proof of Theorem 3.5, and replacing ${\mathcal{M}}_n$ by ${\mathcal{S}}_n\left(\mathbb{F}\right)$, replacing the basis (48) of ${\mathcal{M}}_n$ by the following basis of rank 1 projections in ${\mathcal{S}}_n\left(\mathbb{F}\right)$:

$$\left\{E_{\mathit{ii}}:1\le i\le n\right\}\cup \left\{\frac{1}{\sqrt{2}}\left(E_{\mathit{ii}}+E_{\mathit{jj}}+E_{\mathit{ij}}+E_{\mathit{ji}}\right):1\le i<j\le n\right\},$$E87

and replacing the usage of Theorem 3.4 by that of Theorem 5.2, we can prove the case $k\ge 2$, and for $k<0$, we can get the corresponding equalities up to (60).

Define a linear map ${\psi }_1:{\mathcal{S}}_n\left(\mathbb{F}\right)\to {\mathcal{M}}_n\left(\mathbb{F}\right)$ by ${\psi }_1\left(B\right)≔\psi \left(B\right)\psi {\left(I_n\right)}^{-1}$.

When $k=-1$, we get the corresponding equality of (61), so that ${\psi }_1\left(B^2\right)={\psi }_1{\left(B\right)}^2$ for $B\in {\mathcal{S}}_n\left(\mathbb{F}\right)$. Similar to the proof of Theorem 5.2, we get ${\psi }_1\left(B^r\right)={\psi }_1{\left(B\right)}^r$ for all $r\in {\mathbb{Z}}_{+}$. By [26, Theorem 6.5.3], there is an invertible matrix $P\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that ${\psi }_1\left(B\right)={\mathit{PBP}}^{-1}$, so that $\psi \left(B\right)={\mathit{PBP}}^{-1}\psi \left(I_n\right)$ for $B\in {\mathcal{S}}_n\left(\mathbb{F}\right)$. Since $\psi \left(B\right)=\psi {\left(B\right)}^t$, we get

$$\left(P^{-1}\psi \left(I_n\right)P^{-t}\right)B=B\left(P^{-1}\psi \left(I_n\right)P^{-t}\right),B\in {\mathcal{S}}_n\left(\mathbb{F}\right).$$E88

Therefore, $P^{-1}\psi \left(I_n\right)P^{-t}={\mathit{cI}}_n$ for certain $c\in \mathbb{F}\backslash \left\{0\right\}$, so that $\psi \left(B\right)={\mathit{\text{cPBP}}}^t$ for all $B\in {\mathcal{S}}_n\left(\mathbb{F}\right)$. Consequently, we get (85). The remaining claims are obvious.

When $k<-1$, using analogous argument as in the proof of $k<-1$ case of Theorem 3.5 and applying Theorem 5.2, we can get (86).

6. $k$-power linear preservers and trace of power-product preservers on ${\mathcal{P}}_n$ and ${\mathcal{P}}_n\left(\mathbb{R}\right)$

In this section, we will determine $k$-power linear preservers and trace of power-product preservers on maps ${\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ (resp. ${\mathcal{P}}_n\left(\mathbb{R}\right)\to \overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}$). Properties of such maps can be applied to maps ${\mathcal{P}}_n\to {\mathcal{P}}_n$ and $\overline{{\mathcal{P}}_n}\to \overline{{\mathcal{P}}_n}$ (resp. ${\mathcal{P}}_n\left(\mathbb{R}\right)\to {\mathcal{P}}_n\left(\mathbb{R}\right)$ and $\overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}\to \overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}$).

6.1 $k$-power linear preservers on ${\mathcal{P}}_n$ and ${\mathcal{P}}_n\left(\mathbb{R}\right)$

Theorem 6.1. Fix $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. A nonzero linear map $\psi :{\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ (resp. $\psi :{\mathcal{P}}_n\left(\mathbb{R}\right)\to \overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}$) satisfies that

$$\psi \left(A^k\right)=\psi {\left(A\right)}^k$$E89

on an open neighborhood of $I_n$ in ${\mathcal{P}}_n$ (resp. ${\mathcal{P}}_n\left(\mathbb{R}\right)$) if and only if there is a unitary (resp. real orthogonal) matrix $U\in M_n$ such that

$$\psi \left(A\right)=U^{\ast }\mathit{AU},A\in {\mathcal{P}}_n;\mathit{\text{or}}\psi \left(A\right)=U^{\ast }{A}^{t}U,A\in {\mathcal{P}}_n.$$E90

Proof. We prove the case $\psi :{\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$. The sufficient part is obvious. About the necessary part, the nonzero linear map $\psi :{\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ can be easily extended to a linear map $\tilde{\psi }:{\mathcal{H}}_n\to {\mathcal{H}}_n$ that satisfies $\tilde{\psi }\left(A^k\right)=\tilde{\psi }{\left(A\right)}^k$ on an open neighborhood of $I_n$. By Theorem 4.1, we immediately get (90).

The case $\psi :{\mathcal{P}}_n\left(\mathbb{R}\right)\to \overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}$ can be similarly proved using Theorem 5.2.

6.2 Trace of powered product preservers on ${\mathcal{P}}_n$ and ${\mathcal{P}}_n\left(\mathbb{R}\right)$

Now consider the maps ${\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ (resp. ${\mathcal{P}}_n\left(\mathbb{R}\right)\to \overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}$) that preserve trace of powered products. Unlike ${\mathcal{M}}_n$ and ${\mathcal{H}}_n$, the set ${\mathcal{P}}_n$ (resp. ${\mathcal{P}}_n\left(\mathbb{R}\right)$) is not a vector space. The trace of powered product preservers of two maps have the following forms.

Theorem 6.2 (Huang, Tsai [18]). Let $a,b,c,d\in \mathbb{R}\backslash \left\{0\right\}$. Two maps $\varphi ,\psi :{\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ satisfy

$$\mathrm{tr}\left(\varphi {\left(A\right)}^a\psi {\left(B\right)}^b\right)=\mathrm{tr}\left(A^c{B}^d\right),A,B\in {\mathcal{P}}_n,$$E91

if and only if there exists an invertible $P\in {\mathcal{M}}_n$ such that

$$\left(\begin{array}{c}\varphi \left(A\right)={\left(P^{\ast }{A}^{c}P\right)}^{1/a}\\ \psi \left(B\right)={\left(P^{-1}{B}^d{P}^{-\ast }\right)}^{1/b}\end{array}\right.\mathit{\text{or}}\left(\begin{array}{c}\varphi \left(A\right)={\left[P^{\ast }{\left(A^t\right)}^{c}P\right]}^{1/a}\\ \psi \left(B\right)={\left[P^{-1}{\left(B^t\right)}^d{P}^{-\ast }\right]}^{1/b}\end{array}\right.A,B\in {\mathcal{P}}_n.$$E92

Theorem 6.3 (Huang, Tsai [18]). Given an integer $m\ge 3$ and real numbers ${\alpha }_1,\dots ,{\alpha }_m,{\beta }_1,\dots ,{\beta }_m\in \mathbb{R}\backslash \left\{0\right\}$, maps ${\varphi }_i:{\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ ($i=1,\dots ,m$) satisfy that

$$\mathrm{tr}\left({\varphi }_1{\left(A_1\right)}^{{\alpha }_1}\cdots {\varphi }_m{\left(A_m\right)}^{{\alpha }_m}\right)=\mathrm{tr}\left(A_1^{{\beta }_1}\cdots A_m^{{\beta }_1}\right),A_1,\dots ,A_m\in {\mathcal{P}}_n,$$E93

if and only if they have the following forms for certain $c_1,\dots ,c_m\in {\mathbb{R}}_{+}$ with $c_1\cdots c_m=1$:

1. When $m$ is odd, there exists a unitary matrix $U\in {\mathcal{M}}_n$ such that for $i=1,\dots ,m$:

   $${\varphi }_i\left(A\right)=c_i^{1/{\alpha }_i}{U}^{\ast }{A}^{{\beta }_i/{\alpha }_i}U,A\in {\mathcal{P}}_n.$$E94

2. When $m$ is even, there exists an invertible $M\in {\mathcal{M}}_n$ such that for $i=1,\dots ,m$:

   $${\varphi }_i\left(A\right)=\left(\begin{array}{cc}{c}_i^{1/{\alpha }_i}{\left(M^{\ast }{A}^{{\beta }_i}M\right)}^{1/{\alpha }_i},& i\mathit{\text{is}}\mathit{odd},\\ c_i^{1/{\alpha }_i}{\left(M^{-1}{A}^{{\beta }_i}{M}^{-\ast }\right)}^{1/{\alpha }_i},& i\mathit{\text{is even}},\end{array}\right.A\in {\mathcal{P}}_n.$$E95

   Both Theorems 6.2 and 6.3 can be analogously extended to maps ${\mathcal{P}}_n\left(\mathbb{R}\right)\to \overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}$ without difficulties.

   Theorem 6.2 determines maps $\varphi ,\psi :{\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ that satisfy (91) throughout their domain. If we only assume the equality (91) for $\left(A,B\right)$ in certain subset of ${\mathcal{P}}_n\times {\mathcal{P}}_n$ and assume certain linearity of $\varphi$ and $\psi$, then $\varphi$ and $\psi$ may have slightly different forms. We determine the case $a=c=1$ and $b=d=k\in \mathbb{Z}\backslash \left\{0\right\}$ here.

   Theorem 6.4. Let $k\in \mathbb{Z}\backslash \left\{0\right\}$. Let $S$ be an open neighborhood of $I_n$ in ${\mathcal{P}}_n$. Two maps $\varphi ,\psi :{\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ satisfy

   $$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left({\mathit{AB}}^k\right),$$E96

3. for all $A,B\in {\mathcal{P}}_n$, or

4. for all $A\in S,B\in {\mathcal{P}}_n$, and $\varphi$ is linear,

if and only if there exists an invertible $P\in {\mathcal{M}}_n$ such that

$$\left(\begin{array}{l}\varphi \left(A\right)=P^{\ast }\mathit{AP}\\ \psi \left(B\right)={\left(P^{-1}{B}^k{P}^{-\ast }\right)}^{1/k}\end{array}\right.\mathit{\text{or}}\left(\begin{array}{l}\varphi \left(A\right)=P^{\ast }{A}^{t}P\\ \psi \left(B\right)={\left[P^{-1}{\left(B^t\right)}^k{P}^{-\ast }\right]}^{1/k}\end{array}\right.A,B\in {\mathcal{P}}_n.$$E97

The maps $\varphi$ and $\psi$ satisfy (96)

1. for all $A\in {\mathcal{P}}_n,B\in S$, and $\psi$ is linear, or

2. for all $A,B\in S$ and both $\varphi$ and $\psi$ are linear,

if and only if when $k\in \left\{-1,1\right\}$, $\varphi$ and $\psi$ take the form (97), and when $k\in \mathbb{Z}\backslash \left\{-\mathrm{1,0,1}\right\}$, there exist a unitary matrix $U\in {\mathcal{M}}_n$ and $c\in {\mathbb{R}}^{+}$ such that

$$\left(\begin{array}{c}\varphi \left(A\right)=c^{-k}{U}^{\ast }\mathit{AU}\\ \psi \left(B\right)={\mathit{cU}}^{\ast }\mathit{BU}\end{array}\right.\mathit{\text{or}}\left(\begin{array}{c}\varphi \left(A\right)=c^{-k}{U}^{\ast }{A}^{t}U\\ \psi \left(B\right)={\mathit{cU}}^{\ast }{B}^{t}U\end{array}\right.A,B\in {\mathcal{P}}_n.$$E98

Proof. It suffices to prove the necessary part.

The case of assumption (1) has been proved by Theorem 6.2.

Similar to the proof of Theorem 3.5, assumption (2) implies assumption (1); assumption (4) implies assumption (3). It remains to prove the case with assumption (3).

When $k=1$, assumption (3) is analogous to assumption (2), and we get (97).

Suppose $k\in \mathbb{Z}\backslash \left\{1,0\right\}$. Let ${\psi }_1:{\mathcal{P}}_n\to \overline{{\mathcal{P}}_n}$ be defined by ${\psi }_1\left(B\right)≔\psi {\left(B^{1/k}\right)}^k$. Let ${\mathcal{S}}_1≔\left\{B\in {\mathcal{P}}_n:B^{1/k}\in S\right\}.$ Then (97) with assumption (2) becomes

$$\mathrm{tr}\left(\varphi \left(A\right){\psi }_1\left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right),A\in {\mathcal{P}}_n,B\in {\mathcal{S}}_1.$$E99

Let $\tilde{\psi }:{\mathcal{H}}_n\to {\mathcal{H}}_n$ be the linear extension of $\psi$. By Theorem 2.2, $\varphi$ can be extended to a linear bijection $\tilde{\varphi }:{\mathcal{H}}_n\to {\mathcal{H}}_n$ such that

$$\mathrm{tr}\left(\tilde{\varphi }\left(A\right)\tilde{\psi }{\left(B\right)}^k\right)=\mathrm{tr}\left(\tilde{\varphi }\left(A\right){\psi }_1\left(B^k\right)\right)=\mathrm{tr}\left({\mathit{AB}}^k\right),A\in {\mathcal{H}}_n,B\in S.$$E100

By Theorem 4.2 and taking into account the ranges of $\varphi$ and $\psi$, we see that when $k=-1$, $\varphi$ and $\psi$ take the form of (97), and when $k\in \mathbb{Z}\backslash \left\{-\mathrm{1,0,1}\right\}$, $\varphi$ and $\psi$ take the form of (98).

Theorem 6.4 has counterpart results for $\varphi ,\psi :{\mathcal{P}}_n\left(\mathbb{R}\right)\to \overline{{\mathcal{P}}_n\left(\mathbb{R}\right)}$ and the proof is analogous using Theorem 5.3 instead of Theorem 4.2.

7. $k$-power linear preservers and trace of power-product preservers on ${\mathcal{D}}_n$ and ${\mathcal{D}}_n\left(\mathbb{R}\right)$

Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Define the function $diag:{\mathbb{F}}^n\to {\mathcal{D}}_n\left(\mathbb{F}\right)$ to be the linear bijection that sends each ${\left(c_1,\cdots ,c_n\right)}^t$ to the diagonal matrix with $c_1,\dots ,c_n$ (in order) as the diagonal entries. Define ${diag}^{-1}:{\mathcal{D}}_n\left(\mathbb{F}\right)\to {\mathbb{F}}^n$ the inverse map of $diag$.

With the settings, every linear map $\psi :{\mathcal{D}}_n\left(\mathbb{F}\right)\to {\mathcal{D}}_n\left(\mathbb{F}\right)$ uniquely corresponds to a matrix $L_{\psi }\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\psi \left(A\right)=diag\left(L_{\psi }{diag}^{-1}\left(A\right)\right),A\in {\mathcal{D}}_n\left(\mathbb{F}\right).$$E101

7.1 $k$-power linear preservers on ${\mathcal{D}}_n$ and ${\mathcal{D}}_n\left(\mathbb{R}\right)$

We define the linear functionals $f_i:{\mathcal{D}}_n\left(\mathbb{F}\right)\to \mathbb{F}$ $\left(i=0,1,\dots ,n\right)$, such that for each $A=diag\left(a_1,\dots ,a_n\right)\in {\mathcal{D}}_n\left(\mathbb{F}\right)$,

$$f_0\left(A\right)=0;f_i\left(A\right)=a_i,i=1,\dots ,n.$$E102

Theorem 7.1. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. Let $S$ be an open neighborhood of $I_n$ in ${\mathcal{D}}_n\left(\mathbb{F}\right)$. A linear map $\psi :{\mathcal{D}}_n\left(\mathbb{F}\right)\to {\mathcal{D}}_n\left(\mathbb{F}\right)$ satisfies that

$$\psi \left(A^k\right)=\psi {\left(A\right)}^k,A\in S,$$E103

if and only if

$$\psi \left(A\right)=\psi \left(I_n\right)diag\left(f_{p\left(1\right)}\left(A\right),\dots ,f_{p\left(n\right)}\left(A\right)\right),A\in {\mathcal{D}}_n\left(\mathbb{F}\right),$$E104

in which $\psi {\left(I_n\right)}^k=\psi \left(I_n\right)$ and $p:\left\{1,\dots ,n\right\}\to \left\{0,1,\dots ,n\right\}$ is a function such that $p\left(i\right)\ne 0$ when $k<0$ for $i=1,\dots ,n$. In particular, a linear bijection $\psi :{\mathcal{D}}_n\left(\mathbb{F}\right)\to {\mathcal{D}}_n\left(\mathbb{F}\right)$ satisfies (103) if and only if there is a diagonal matrix $C\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ with $C^{k-1}=I_n$ and a permutation matrix $P\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\psi \left(A\right)={\mathit{\text{PCAP}}}^{-1},A\in {\mathcal{D}}_n\left(\mathbb{F}\right).$$E105

Proof. For every $A=diag\left(a_1,\dots ,a_n\right)\in {\mathcal{D}}_n\left(\mathbb{F}\right)$, when $x\in \mathbb{F}$ is sufficiently close to $0$, we have $I_n+\mathit{xA}\in S$ and the power series of ${\left(I_n+\mathit{xA}\right)}^k$ converges, so that $\psi \left({\left(I_n+\mathit{xA}\right)}^k\right)=\psi {\left(I_n+\mathit{xA}\right)}^k$.

$$\psi \left({\left(I_n+\mathit{xA}\right)}^k\right)=\psi \left(I_n\right)+\mathit{xk\psi }\left(A\right)+x^2\frac{k\left(k-1\right)}{2}\psi \left(A^2\right)+\cdots$$E106

$$\psi {\left(I_n+\mathit{xA}\right)}^k=\psi \left(I_n\right)+\mathit{xk\psi }{\left(I_n\right)}^{k-1}\psi \left(A\right)+x^2\frac{k\left(k-1\right)}{2}\psi {\left(I_n\right)}^{k-2}\psi {\left(A\right)}^2+\cdots$$E107

So for all $A\in {\mathcal{D}}_n\left(\mathbb{F}\right)$:

$$\psi \left(A\right)=\psi {\left(I_n\right)}^{k-1}\psi \left(A\right),$$E108

$$\psi \left(A^2\right)=\psi {\left(I_n\right)}^{k-2}\psi {\left(A\right)}^2.$$E109

The linear map ${\psi }_1\left(A\right)≔\psi {\left(I_n\right)}^{k-2}\psi \left(A\right)$ satisfies that

$${\psi }_1\left(A^2\right)={\psi }_1{\left(A\right)}^2,A\in {\mathcal{D}}_n\left(\mathbb{F}\right).$$E110

By (101), let $L_{{\psi }_1}=\left({\mathrm{\ell }}_{\mathit{ij}}\right)\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that ${diag}^{-1}\left({\psi }_1\left(A\right)\right)=L_{{\psi }_1}\left({diag}^{-1}\left(A\right)\right)$ for $A\in {\mathcal{D}}_n\left(\mathbb{F}\right)$. Then (110) implies that for all $A=diag\left(a_1,\dots ,a_n\right)\in {\mathcal{D}}_n\left(\mathbb{F}\right)$:

$$\sum _{j=1}^n{\mathrm{\ell }}_{\mathit{ij}}{a}_j^2={\left(\sum _{j=1}^n{\mathrm{\ell }}_{\mathit{ij}}{a}_j\right)}^2,i=1,2,\dots ,n.$$E111

Therefore, each row of $L_{{\psi }_1}$ has at most one nonzero entry and each nonzero entry must be 1. We get

$${\psi }_1\left(A\right)=diag\left(f_{p\left(1\right)}\left(A\right),\dots ,f_{p\left(n\right)}\left(A\right)\right)$$E112

in which $p:\left\{1,\dots ,n\right\}\to \left\{0,1,\dots ,n\right\}$ is a function. Suppose $\psi \left(I_n\right)=diag\left({\lambda }_1,\dots ,{\lambda }_n\right)$. Then (108) implies that $\psi \left(A\right)=\psi \left(I_n\right){\psi }_1\left(A\right)$ has the form (104). Obviously, $\psi {\left(I_n\right)}^k=\psi \left(I_n\right)$ and when $k<0$, each $p\left(i\right)\ne 0$ for $i=1,\dots ,n$. Moreover, when $\psi$ is a linear bijection, (112) shows that ${\psi }_1\left(A\right)={\mathit{PAP}}^{-1}$ for a permutation matrix $P$. (105) can be easily derived.

7.2 Trace of power-product preservers on ${\mathcal{D}}_n$ and ${\mathcal{D}}_n\left(\mathbb{R}\right)$

In [18], we show that two maps $\varphi ,\psi :{\mathcal{D}}_n\left(\mathbb{F}\right)\to {\mathcal{D}}_n\left(\mathbb{F}\right)$ satisfy $\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathrm{tr}\left(\mathit{AB}\right)$ for $A,B\in {\mathcal{D}}_n\left(\mathbb{F}\right)$ if and only if there exists an invertible $N\in {\mathcal{M}}_n\left(\mathbb{F}\right)$ such that

$$\varphi \left(A\right)=diag\left(N{diag}^{-1}\left(A\right)\right),\psi \left(B\right)=diag\left(N^{-t}{diag}^{-1}\left(B\right)\right),A,B\in {\mathcal{D}}_n\left(\mathbb{F}\right).$$E113

When $m\ge 3$, the maps ${\varphi }_1,\dots ,{\varphi }_m:{\mathcal{D}}_n\left(\mathbb{F}\right)\to {\mathcal{D}}_n\left(\mathbb{F}\right)$ satisfying $\mathit{tr}\left({\varphi }_1\left(A_1\right)\cdots {\varphi }_m\left(A_m\right)\right)=\mathit{tr}\left(A_1\cdots A_m\right)$ for $A_1,\dots ,A_m\in {\mathcal{D}}_n\left(\mathbb{F}\right)$ are also determined in [18].

Next we consider the trace of power-product preserver on ${\mathcal{D}}_n\left(\mathbb{F}\right)$.

Theorem 7.2. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. Let $S$ be an open neighborhood of $I_n$ in ${\mathcal{D}}_n\left(\mathbb{F}\right)$. Two maps $\varphi ,\psi :{\mathcal{D}}_n\left(\mathbb{F}\right)\to {\mathcal{D}}_n\left(\mathbb{F}\right)$ satisfy that

$$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left({\mathit{AB}}^k\right),$$E114

1. for all $A\in {\mathcal{D}}_n\left(\mathbb{F}\right),B\in S$, and $\psi$ is linear, or

2. for all $A,B\in S$ and both $\varphi$ and $\psi$ are linear,

if and only if there exist an invertible diagonal matrix $C\in {\mathcal{D}}_n\left(\mathbb{F}\right)$ and a permutation matrix $P\in {\mathcal{M}}_n\left(F\right)$ such that

$$\varphi \left(A\right)={\mathit{PC}}^{-k}{\mathit{AP}}^{-1},\psi \left(B\right)={\mathit{\text{PCBP}}}^{-1},A,B\in {\mathcal{D}}_n\left(\mathbb{F}\right).$$E115

Proof. Assumption (2) leads to assumption (1) (cf. the proof of Theorem 3.5). We prove the theorem under assumption (1).

For every $B\in {\mathcal{D}}_n\left(\mathbb{F}\right)$, $I_n+\mathit{xB}\in S$ and the power series of ${\left(I_n+\mathit{xB}\right)}^k$ converges when $x\in \mathbb{F}$ is sufficiently close to $0$, so that

$$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(I_n+\mathit{xB}\right)}^k\right)=\mathrm{tr}\left(A{\left(I_n+\mathit{xB}\right)}^k\right)$$E116

Comparing degree one terms and degree two terms in the power series of the above equality, respectively, we get the following equalities for $A,B\in {\mathcal{D}}_n\left(\mathbb{F}\right)$:

$$\mathrm{tr}\left(\varphi \left(A\right)\psi \left(B\right)\psi {\left(I_n\right)}^{k-1}\right)=\mathrm{tr}\left(\mathit{AB}\right),$$E117

$$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^2\psi {\left(I_n\right)}^{k-2}\right)=\mathrm{tr}\left({\mathit{AB}}^2\right).$$E118

Applying Theorem 2.2 to (117), $\psi \left(I_n\right)$ is invertible and both $\varphi$ and $\psi$ are linear bijections. (117) and (119) imply that $\psi \left(B^2\right)\psi {\left(I_n\right)}^{k-1}=\psi {\left(B\right)}^2\psi {\left(I_n\right)}^{k-2}$. Let ${\psi }_1\left(B\right)≔\psi \left(B\right)\psi {\left(I_n\right)}^{-1}$. Then ${\psi }_1\left(B^2\right)={\psi }_1{\left(B\right)}^2$ for $B\in {\mathcal{D}}_n\left(\mathbb{F}\right)$. By Theorem 7.1 and ${\psi }_1\left(I_n\right)=I_n$, there exists a permutation matrix $P\in {\mathcal{M}}_n\left(F\right)$ such that ${\psi }_1\left(B\right)={\mathit{PBP}}^{-1}$ for $B\in {\mathcal{D}}_n\left(\mathbb{F}\right)$. So $\psi \left(B\right)=\psi \left(I_n\right){\mathit{PBP}}^{-1}={\mathit{\text{PCBP}}}^{-1}$ for $C≔P^{-1}\psi \left(I_n\right)P\in {\mathcal{D}}_n\left(\mathbb{F}\right)$. Then (114) implies (115).

8. $k$-power injective linear preservers and trace of power-product preservers on ${\mathcal{T}}_n$ and ${\mathcal{T}}_n\left(\mathbb{R}\right)$

8.1 $k$-power preservers on ${\mathcal{T}}_n\left(\mathbb{F}\right)$

The characterization of injective linear $k$-power preserver on ${\mathcal{T}}_n\left(\mathbb{F}\right)$ can be derived from Cao and Zhang’s characterization of injective additive $k$-power preserver on ${\mathcal{T}}_n\left(\mathbb{F}\right)$ ([12] or [[6], Theorem 6.5.2]).

Theorem 8.1 (Cao and Zhang [12]). Let $k\ge 2$ and $n\ge 3$. Let $\mathbb{F}$ be a field with $\mathit{char}\left(\mathbb{F}\right)=0$ or $\mathit{char}\left(\mathbb{F}\right)>k$. Then $\psi :{\mathcal{T}}_n\left(\mathbb{F}\right)\mapsto {\mathcal{T}}_n\left(\mathbb{F}\right)$ is an injective linear map such that $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A\in {\mathcal{T}}_n\left(\mathbb{F}\right)$ if and only if there exists a $\left(k-1\right)$th root of unity $\lambda$ and an invertible matrix $P\in {\mathcal{T}}_n\left(\mathbb{F}\right)$ such that

$$\psi \left(A\right)=\lambda {\mathit{PAP}}^{-1},A\in {\mathcal{T}}_n\left(\mathbb{F}\right),\mathit{\text{or}}$$E119

$$\psi \left(A\right)=\lambda {\mathit{PA}}^{-}{P}^{-1},A\in {\mathcal{T}}_n\left(\mathbb{F}\right),$$E120

where $A^{-}=\left(a_{n+1-j,n+1-i}\right)$ if $A=\left(a_{\mathit{ij}}\right)$.

Example 8.2. When $n=2$, the injective linear maps that satisfy $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for $A\in {\mathcal{T}}_2\left(\mathbb{F}\right)$ send $A=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ 0& a_{22}\end{array}\right)$ to the following $\psi \left(A\right)$:

$$\lambda \left(\begin{array}{cc}{a}_{11}& {\mathit{ca}}_{12}\\ 0& a_{22}\end{array}\right),\lambda \left(\begin{array}{cc}{a}_{22}& {\mathit{ca}}_{12}\\ 0& a_{11}\end{array}\right),$$E121

in which ${\lambda }^{k-1}=1$ and $c\in \mathbb{F}\backslash \left\{0\right\}.$.

Example 8.3. Theorem 8.1 does not hold if $\psi$ is not assumed to be injective. Let $n=3$ and suppose $\psi :{\mathcal{T}}_3\left(\mathbb{F}\right)\to {\mathcal{T}}_3\left(\mathbb{F}\right)$ is a linear map that sends $A={\left(\begin{array}{c}{a}_{\mathit{ij}}\end{array}\right)}_{3\times 3}\in {\mathcal{T}}_3\left(\mathbb{F}\right)$ to one of the following $\psi \left(A\right)$ ($c,d\in \mathbb{F}$):

$$\left(\begin{array}{ccc}{a}_{11}& {\mathit{ca}}_{12}& 0\\ 0& a_{22}& {\mathit{da}}_{23}\\ 0& 0& a_{33}\end{array}\right),\left(\begin{array}{ccc}{a}_{33}& 0& 0\\ 0& a_{11}& 0\\ 0& 0& a_{22}\end{array}\right),\left(\begin{array}{ccc}{a}_{22}& 0& {\mathit{ca}}_{12}\\ 0& 0& 0\\ 0& 0& a_{11}\end{array}\right).$$E122

Then each $\psi$ satisfies that $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for every positive integer $k$ but it is not of the forms in Theorem 8.1.

We extend Theorem 8.1 to the following result that includes negative $k$-powers and that only assumes $k$-power preserving in a neighborhood of $I_n$.

Theorem 8.4. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let integers $k\ne 0,1$ and $n\ge 3$. Suppose that $\psi :{\mathcal{T}}_n\left(\mathbb{F}\right)\to {\mathcal{T}}_n\left(\mathbb{F}\right)$ is an injective linear map such that $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ for all $A$ in an open neighborhood of $I_n$ in ${\mathcal{T}}_n\left(\mathbb{F}\right)$ consisting of invertible matrices. Then there exist $\lambda \in \mathbb{F}$ with ${\lambda }^{k-1}=1$ and an invertible matrix $P\in {\mathcal{T}}_n\left(\mathbb{F}\right)$ such that

$$\psi \left(A\right)=\lambda {\mathit{PAP}}^{-1},A\in {\mathcal{T}}_n\left(\mathbb{F}\right),\mathit{\text{or}}$$E123

$$\psi \left(A\right)=\lambda {\mathit{PA}}^{-}{P}^{-1},A\in {\mathcal{T}}_n\left(\mathbb{F}\right).$$E124

where $A^{-}=\left(a_{n+1-j,n+1-i}\right)=J_n{A}^t{J}_n$ if $A=\left(a_{\mathit{ij}}\right)$, $J_n$ is the anti-diagonal identity.

Proof. Obviously $\psi$ is a linear bijection. Follow the same process in the proof of Theorem 3.4. In both $k\ge 2$ and $k<0$ cases we have $\psi \left(I_n\right)$ commutes with the range of $\psi$, so that $\psi \left(I_n\right)=\lambda I_n$ for $\lambda \in \mathbb{F}$ and ${\lambda }^{k-1}=1.$ Moreover, let ${\psi }_1\left(A\right)≔\psi {\left(I_n\right)}^{-1}\psi \left(A\right)$, then ${\psi }_1$ is injective linear and ${\psi }_1\left(A^2\right)={\psi }_1{\left(A\right)}^2$ for $A\in {\mathcal{T}}_n\left(\mathbb{F}\right).$ Theorem 8.1 shows that ${\psi }_1\left(A\right)={\mathit{PAP}}^{-1}$ or ${\psi }_1\left(A\right)={\mathit{PA}}^{-}{P}^{-1}$ for certain invertible $P\in {\mathcal{T}}_n\left(\mathbb{F}\right).$ It leads to (123) and (124).

8.2 Trace of power-product preservers on ${\mathcal{T}}_n$ and ${\mathcal{T}}_n\left(\mathbb{R}\right)$

Theorem 2.2 or Corollary 2.3 does not work for maps on ${\mathcal{T}}_n\left(\mathbb{F}\right)$. However, the following trace preserving result can be easily derived from Theorem 7.2. We have ${\mathcal{T}}_n\left(\mathbb{F}\right)={\mathcal{D}}_n\left(\mathbb{F}\right)\oplus {\mathcal{N}}_n\left(\mathbb{F}\right)$. Let $D\left(A\right)$ denote the diagonal matrix that takes the diagonal of $A\in {\mathcal{T}}_n\left(\mathbb{F}\right)$.

Theorem 8.5. Let $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let $k\in \mathbb{Z}\backslash \left\{0,1\right\}$. Let $S$ be an open neighborhood of $I_n$ in ${\mathcal{T}}_n\left(\mathbb{F}\right)$ consisting of invertible matrices. Then two maps $\varphi ,\psi :{\mathcal{T}}_n\left(\mathbb{F}\right)\to {\mathcal{T}}_n\left(\mathbb{F}\right)$ satisfy that

$$\mathrm{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathrm{tr}\left({\mathit{AB}}^k\right),$$E125

1. for all $A\in {\mathcal{T}}_n\left(\mathbb{F}\right),B\in S$, and $\psi$ is linear, or

2. for all $A,B\in S$ and both $\varphi$ and $\psi$ are linear,

if and only if $\varphi$ and $\psi$ send ${\mathcal{N}}_n\left(\mathbb{F}\right)$ to ${\mathcal{N}}_n\left(\mathbb{F}\right)$, ${\left.\left(D\circ \varphi \right)\right|}_{{\mathcal{D}}_n\left(\mathbb{F}\right)}$ and ${\left.\left(D\circ \psi \right)\right|}_{{\mathcal{D}}_n\left(\mathbb{F}\right)}$ are linear bijections characterized by (115) in Theorem 7.2, and $D\circ \varphi =D\circ \varphi \circ D$.

Proof. The sufficient part is easy to verify. We prove the necessary part here. Let ${\left.{\varphi }^{\prime }≔\left(D\circ \varphi \right)\right|}_{{\mathcal{D}}_n\left(\mathbb{F}\right)}$ and ${\left.{\psi }^{\prime }≔\left(D\circ \psi \right)\right|}_{{\mathcal{D}}_n\left(\mathbb{F}\right)}$. Then ${\varphi }^{\prime },{\psi }^{\prime }:{\mathcal{D}}_n\left(\mathbb{F}\right)\to {\mathcal{D}}_n\left(\mathbb{F}\right)$ satisfy $\mathit{tr}\left({\varphi }^{\prime }\left(A\right){\psi }^{\prime }{\left(B\right)}^k\right)=\mathit{tr}\left({\mathit{AB}}^k\right)$ for $A,B\in {\mathcal{D}}_n\left(\mathbb{F}\right)$. So they are characterized by (115). The bijectivity of ${\varphi }^{\prime }$ and ${\psi }^{\prime }$ implies that $\varphi$ and $\psi$ must send ${\mathcal{N}}_n\left(\mathbb{F}\right)$ to ${\mathcal{N}}_n\left(\mathbb{F}\right)$ in order to satisfy (125). Moreover, $\varphi$ should send matrices with same diagonal to matrices with same diagonal, which implies that $D\circ \varphi =D\circ \varphi \circ D$.

9. Conclusion

We characterize linear maps $\psi :V\to V$ that satisfy $\psi \left(A^k\right)=\psi {\left(A\right)}^k$ on an open neighborhood $S$ of $I_n$ in $V$, where $k\in \mathbb{Z}\backslash \left\{0,1\right\}$ and $V$ is the set of $n\times n$ general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices, over the complex or real field. The characterizations extend the existing results of linear $k$-power preservers on the spaces of general matrices, symmetric matrices, and upper triangular matrices.

Applying the above results, we determine the maps $\varphi ,\psi :V\to V$ on the preceding sets $V$ that satisfy $\mathit{tr}\left(\varphi \left(A\right)\psi {\left(B\right)}^k\right)=\mathit{tr}\left({\mathit{AB}}^k\right)$

1. for all $A\in V$, $B\in S$, and $\psi$ is linear, or

2. for all $A,B\in S$ and both $\varphi$ and $\psi$ are linear.

These results, together with Theorem 2.2 about maps satisfying $\mathit{tr}\left(\varphi \left(A\right)\psi \left(B\right)\right)=\mathit{tr}\left(\mathit{AB}\right)$ and the characterizations of maps ${\varphi }_1,\cdots ,{\varphi }_m:V\to V$ ($m\ge 3$) satisfying $\mathit{tr}\left({\varphi }_1\left(A_1\right)\cdots {\varphi }_m\left(A_m\right)\right)=\mathit{tr}\left(A_1\cdots A_m\right)$ in [18], make a comprehensive picture of the preservers of trace of matrix products in the related matrix spaces and sets. Our results can be interpreted as inner product preservers when $V$ is close under conjugate transpose, in which wide applications are found.

There are a few prospective directions to further the researches.

First, for a polynomial or an analytic function $f\left(x\right)$ and a matrix set $V$, we can consider “local” linear $f$-preservers, that is, linear operators $\psi :V\to V$ that satisfy $\psi \left(f\left(A\right)\right)=f\left(\psi \left(A\right)\right)$ on an open subset $S$ of $V$. A linear $f$-preserver $\psi$ on $S$ also preserves matrices annihilated by $f$ on $S$, that is, $f\left(A\right)=0$ ($A\in S$) implies $f\left(\psi \left(A\right)\right)=0$. When $S=V$ is $M_n$, $B\left(H\right)$, or some operator algebras, extensive studies have been done on operators preserving elements annihilated by a polynomial $f$; for examples, the results on $M_n$ by R. Howard in [19], by P. Botta, S. Pierce, and W. Watkins in [20], and by C.-K. Li and S. Pierce in [21], on $B\left(H\right)$ by P. Šemrl [22], on linear maps $\psi :B\left(H\right)\to B\left(K\right)$ by Z. Bai and J. Hou in [23], and on some operator algebras by J. Hou and S. Hou in [24]. We may further explore linear $f$-preservers for a multivariable function $f\left(x_1,\dots ,x_r\right)$, that is, operator $\psi$ satisfying $\psi \left(f\left(A_1,\dots ,A_r\right)\right)=f\left(\psi \left(A_1\right),\dots ,\psi \left(A_r\right)\right)$. The corresponding annihilator preserver problem has been studied in some special cases, for example, on $M_n$ for homogeneous multilinear polynomials by A. E. Guterman and B. Kuzma in [25].

Second, it is interesting to further investigate maps $\varphi ,\psi :V\to V$ that satisfy $\mathit{tr}\left(f\left(\varphi \left(A\right)\right)g\left(\psi \left(B\right)\right)\right)=\mathit{tr}\left(f\left(A\right)g\left(B\right)\right)$ for some polynomials or analytic functions $f\left(x\right)$ and $g\left(x\right)$. This is equivalent to the inner product preserver problem $⟨f{\left(\varphi \left(A\right)\right)}^{\ast },g\left(\psi \left(B\right)\right)⟩=⟨f{\left(A\right)}^{\ast },g\left(B\right)⟩$ when $V$ is close under conjugate transpose. More generally, given a multivariable function $h\left(x_1,\dots ,x_m\right)$, we can ask what combinations of linear operators ${\varphi }_1,\dots ,{\varphi }_m:V\to V$ satisfy that $\mathit{tr}(h\left({\varphi }_1\left(A_1\right),\dots ,{\varphi }_m\left(A_m\right)\right)=\mathit{tr}\left(h\left(A_1,\dots ,A_m\right)\right)$. The research on this area seems pretty new. No much has been discovered by the authors.