Open access peer-reviewed chapter

Operator Means and Applications

By Pattrawut Chansangiam

Submitted: November 11th 2011Reviewed: April 20th 2012Published: July 11th 2012

DOI: 10.5772/46479

Downloaded: 2250

1. Introduction

The theory of scalar means was developed since the ancient Greek by the Pythagoreans until the last century by many famous mathematicians. See the development of this subject in a survey article [1]. In Pythagorean school, various means are defined via the method of proportions (in fact, they are solutions of certain algebraic equations). The theory of matrix and operator means started from the presence of the notion of parallel sum as a tool for analyzing multi-port electrical networks in engineering; see [2]. Three classical means, namely, arithmetic mean, harmonic mean and geometric mean for matrices and operators are then considered, e.g., in [3], [4], [5], [6], [7]. These means play crucial roles in matrix and operator theory as tools for studying monotonicity and concavity of many interesting maps between algebras of operators; see the original idea in [3]. Another important mean in mathematics, namely the power mean, is considered in [8]. The parallel sum is characterized by certain properties in [9]. The parallel sum and these means share some common properties. This leads naturally to the definitions of the so-called connection and mean in a seminal paper [10]. This class of means cover many in-practice operator means. A major result of Kubo-Ando states that there are one-to-one correspondences between connections, operator monotone functions on the non-negative reals and finite Borel measures on the extended half-line. The mean theoretic approach has many applications in operator inequalities (see more information in Section 8), matrix and operator equations (see e.g. [11], [12]) and operator entropy. The concept of operator entropy plays an important role in mathematical physics. The relative operator entropyis defined in [13] for invertible positive operators A,Bby


In fact, this formula comes from the Kubo-Ando theory–S(·|·)is the connection corresponds to the operator monotone function tlogt. See more information in Chapter IV[14] and its references.

In this chapter, we treat the theory of operator means by weakening the original definition of connection in such a way that the same theory is obtained. Moreover, there is a one-to-one correspondence between connections and finite Borel measures on the unit interval. Each connection can be regarded as a weighed series of weighed harmonic means. Hence, every mean in Kubo-Ando's sense corresponds to a probability Borel measure on the unit interval. Various characterizations of means are obtained; one of them is a usual property of scalar mean, namely, the betweenness property. We provide some new properties of abstract operator connections, involving operator monotonicity and concavity, which include specific operator means as special cases.

For benefits of readers, we provide the development of the theory of operator means. In Section 2, we setup basic notations and state some background about operator monotone functions which play important roles in the theory of operator means. In Section 3, we consider the parallel sum together with its physical interpretation in electrical circuits. The arithmetic mean, the geometric mean and the harmonic mean of positive operators are investigated and characterized in Section 4. The original definition of connection is improved in Section 5 in such a way that the same theory is obtained. In Section 6, several characterizations and examples of Kubo-Ando means are given. We provide some new properties of general operator connections, related to operator monotonicity and concavity, in Section 7. Many operator versions of classical inequalities are obtained via the mean-theoretic approach in Section 8.


2. Preliminaries

Throughout, let B()be the von Neumann algebra of bounded linear operators acting on a Hilbert space . Let B()sabe the real vector space of self-adjoint operators on . Equip B()with a natural partial order as follows. For A,BB()sa, we write ABif B-Ais a positive operator. The notation TB()+or T0means that Tis a positive operator. The case that T0and Tis invertible is denoted by T>0or TB()++. Unless otherwise stated, every limit in B()is taken in the strong-operator topology. Write AnAto indicate that Anconverges strongly to A. If Anis a sequence in B()sa, the expression AnAmeans that Anis a decreasing sequence and AnA. Similarly, AnAtells us that Anis increasing and AnA. We always reserve A,B,C,Dfor positive operators. The set of non-negative real numbers is denoted by +.

Remark 0.1It is important to note that if Anis a decreasing sequence in B()sasuch that AnA, then AnAif and only if Anx,xAx,xfor all x. Note first that this sequence is convergent by the order completeness of B(). For the sufficiency, if x, then


and hence (An-A)x0.

The spectrum of TB()is defined by


Then Sp(T)is a nonempty compact Hausdorff space. Denote by C(Sp(T))the C*-algebra of continuous functions from Sp(T)to.LetTB(H)beanormaloperatorandz:Sp(T)the inclusion. Then there exists a unique unital *-homomorphism φ:C(Sp(T))B()such that φ(z)=T, i.e.,

  1. φis linear

  2. φ(fg)=φ(f)φ(g)for all f,gC(Sp(T))

  3. φ(f¯)=(φ(f))*for all fC(Sp(T))

  4. φ(1)=I.

Moreover, φis isometric. We call the unique isometric *-homomorphism which sends fC(Sp(T))to φ(f)B()the continuous functional calculusof T. We write f(T)for φ(f).

Example 0.2

  1. If f(t)=a0+a1t++antn, then f(T)=a0I+a1T++anTn.

  2. If f(t)=t¯, then f(T)=φ(f)=φ(z¯)=φ(z)*=T*

  3. If f(t)=t1/2for t+and T0, then we define T1/2=f(T). Equivalently, T1/2is the unique positive square root of T.

  4. If f(t)=t-1/2for t>0and T>0, then we define T-1/2=f(T). Equivalently, T-1/2=(T1/2)-1=(T-1)1/2.

A continuous real-valued function fon an interval Iis called an operator monotone functionif one of the following equivalent conditions holds:

  1. ABf(A)f(B)for all Hermitian matrices A,Bof all orders whose spectrums are contained in I;

  2. ABf(A)f(B)for all Hermitian operators A,BB()whose spectrums are contained in Iand for an infinite dimensional Hilbert space ;

  3. ABf(A)f(B)for all Hermitian operators A,BB()whose spectrums are contained in Iand for all Hilbert spaces .

This concept is introduced in [15]; see also [14], [16], [17], [18]. Every operator monotone function is always continuously differentiable and monotone increasing. Here are examples of operator monotone functions:

  1. tαt+βon , for α0and β,

  2. t-t-1on (0,),

  3. t(c-t)-1on (a,b), for c(a,b),

  4. tlogton (0,),

  5. t(t-1)/logton +, where 00and 11.

The next result is called the Löwner-Heinz's inequality [15].

Theorem 0.3For A,BB()+and r[0,1], if AB, then ArBr. That is the map ttris an operator monotone function on +for any r[0,1].

A key result about operator monotone functions is that there is a one-to-one correspondence between nonnegative operator monotone functions on +and finite Borel measures on [0,]via integral representations. We give a variation of this result in the next proposition.

Proposition 0.4A continuous function f:++is operator monotone if and only if there exists a finite Borel measure μon [0,1]such that


Here, the weighed harmonic mean !tis defined for a,b>0by


and extended to a,b0by continuity. Moreover, the measure μis unique. Hence, there is a one-to-one correspondence between operator monotone functions on the non-negative reals and finite Borel measures on the unit interval.

Recall that a continuous function f:++is operator monotone if and only if there exists a unique finite Borel measure νon [0,]such that




Consider the Borel measurable function ψ:[0,1][0,],tt1-t.Then, for each x+,


Now, set μ=νψ. Since ψis bijective, there is a one-to-one corresponsence between the finite Borel measures on [0,]of the form νand the finite Borel measures on [0,1]of the form νψ. The map fμis clearly well-defined and bijective.

3. Parallel sum: A notion from electrical networks

In connections with electrical engineering, Anderson and Duffin [2] defined the parallel sumof two positive definite matrices Aand Bby


The impedance of an electrical network can be represented by a positive (semi)definite matrix. If Aand Bare impedance matrices of multi-port networks, then the parallel sum A:Bindicates the total impedance of two electrical networks connected in parallel. This notion plays a crucial role for analyzing multi-port electrical networks because many physical interpretations of electrical circuits can be viewed in a form involving parallel sums. This is a starting point of the study of matrix and operator means. This notion can be extended to invertible positive operators by the same formula.

Lemma 0.5Let A,B,C,D,An,BnB()++for all n.

  1. If AnA, then An-1A-1. If AnA, then An-1A-1.

  2. If ACand BD, then A:BC:D.

  3. If AnAand BnB, then An:BnA:B.

  4. If AnAand BnB, then limAn:Bnexists and does not depend on the choices of An,Bn.

(1) Assume AnA. Then An-1is increasing and, for each x,


(2) Follow from (1).

(3) Let An,BnB()++be such that AnAand BnAwhere A,B>0. Then An-1A-1and Bn-1B-1. So, An-1+Bn-1is an increasing sequence in B()+such that


i.e. An-1+Bn-1A-1+B-1. By (1), we thus have (An-1+Bn-1)-1(A-1+B-1)-1.

(4) Let An,BnB()++be such that AnAand BnB. Then, by (2), An:Bnis a decreasing sequence of positive operators. The order completeness of B()guaruntees the existence of the strong limit of An:Bn. Let An'and Bn'be another sequences such that An'Aand Bn'B. Note that for each n,m, we have AnAn+Am'-Aand BnBn+Bm'-B. Then


Note that as n, An+Am'-AAm'and Bn+Bm'-BBm'. We have that as n,


Hence, limnAn:BnAm':Bm'and limnAn:BnlimmAm':Bm'.By symmetry, limnAn:BnlimmAm':Bm'.

We define the parallel sumof A,B0to be


where the limit is taken in the strong-operator topology.

Lemma 0.6For each x,


First, assume that A,Bare invertible. Then for all x,y,


With y=(A+B)-1Bx, we have


Hence, we have the claim for A,B>0. For A,B0, consider A+ϵIand B+ϵIwhere ϵ0.

Remark 0.7This lemma has a physical interpretation, called the Maxwell's minimum power principle. Recall that a positive operator represents the impedance of a electrical network while the power dissipation of network with impedance Aand current xis the inner product Ax,x. Consider two electrical networks connected in parallel. For a given current input x, the current will divide x=y+z, where yand zare currents of each network, in such a way that the power dissipation is minimum.

Theorem 0.8The parallel sum satisfies

  1. monotonicity: A1A2,B1B2A1:B1A2:B2.

  2. transformer inequality: S*(A:B)S(S*AS):(S*BS)for every SB().

  3. continuity from above: if AnAand BnB, then An:BnA:B.

(1) The monotonicity follows from the formula () and Lemma (2).

(2) For each x,y,zsuch that x=y+z, by the previous lemma,


Again, the previous lemma assures S*(A:B)S(S*AS):(S*BS).

(3) Let Anand Bnbe decreasing sequences in B()+such that AnAand BnB. Then An:Bnis decreasing and A:BAn:Bnfor all n. We have that, by the joint monotonicity of parallel sum, for all ϵ>0


Since An+ϵIA+ϵIand Bn+ϵIB+ϵI, by Lemma 3.1.4(3)we have An:BnA:B.

Remark 0.9The positive operator S*ASrepresents the impedance of a network connected to a transformer. The transformer inequality means that the impedance of parallel connection with transformer first is greater than that with transformer last.

Proposition 0.10The set of positive operators on is a partially ordered commutative semigroup with respect to the parallel sum.

For A,B,C>0, we have (A:B):C=A:(B:C)and A:B=B:A. The continuity from above in Theorem implies that (A:B):C=A:(B:C)and A:B=B:Afor all A,B,C0. The monotonicity of the parallel sum means that the positive operators form a partially ordered semigroup.

Theorem 0.11For A,B,C,D0, we have the series-parallel inequality


In other words, the parallel sum is concave.

For each x,y,zsuch that x=y+z, we have by the previous lemma that


Applying the previous lemma yields (A+B):(C+D)A:C+B:D.

Remark 0.12The ordinary sum of operators represents the total impedance of two networks with series connection while the parallel sum indicates the total impedance of two networks with parallel connection. So, the series-parallel inequality means that the impedance of a series-parallel connection is greater than that of a parallel-series connection.

4. Classical means: arithmetic, harmonic and geometric means

Some desired properties of any object that is called a “mean” Mon B()+should have are given here.

  1. positivity: A,B0M(A,B)0;

  2. monotonicity: AA',BB'M(A,B)M(A',B');

  3. positive homogeneity: M(kA,kB)=kM(A,B)for k+;

  4. transformer inequality: X*M(A,B)XM(X*AX,X*BX)for XB();

  5. congruence invariance: X*M(A,B)X=M(X*AX,X*BX)for invertible XB();

  6. concavity: M(tA+(1-t)B,tA'+(1-t)B')tM(A,A')+(1-t)M(B,B')for t[0,1];

  7. continuity from above: if AnAand BnB, then M(An,Bn)M(A,B);

  8. betweenness: if AB, then AM(A,B)B;

  9. fixed point property: M(A,A)=A.

In order to study matrix or operator means in general, the first step is to consider three classical means in mathematics, namely, arithmetic, geometric and harmonic means.

The arithmetic meanof A,BB()+is defined by


Then the arithmetic mean satisfies the properties (A1)–(A9). In fact, the properties (A5) and (A6) can be replaced by a stronger condition:

X*M(A,B)X=M(X*AX,X*BX)for all XB().

Moreover, the arithmetic mean satisfies

affinity: M(kA+C,kB+C)=kM(A,B)+Cfor k+.

Define the harmonic meanof positive operators A,BB()+by


where AϵA+ϵIand BϵB+ϵI. Then the harmonic mean satisfies the properties (A1)–(A9).

The geometric mean of matrices is defined in [7] and studied in details in [3]. A usage of congruence transformations for treating geometric means is given in [19]. For a given invertible operator CB(), define


Then each ΓCis a linear isomorphism with inverse ΓC-1and is called a congruence transformation. The set of congruence transformations is a group under multiplication. Each congruence transformation preserves positivity, invertibility and, hence, strictly positivity. In fact, ΓCmaps B()+and B()++onto themselves. Note also that ΓCis order-preserving.

Define the geometric meanof A,B>0by


Then A#B>0for A,B>0. This formula comes from two natural requirements: This definition should coincide with the usual geometric mean in +: A#B=(AB)1/2provided that AB=BA. The second condition is that, for any invertible TB(),


The next theorem characterizes the geometric mean of Aand Bin term of the solution of a certain operator equation.

Theorem 0.13For each A,B>0, the Riccati equation ΓX(A-1):=XA-1X=Bhas a unique positive solution, namely, X=A#B.

The direct computation shows that (A#B)A-1(A#B)=B. Suppose there is another positive solution Y0. Then


The uniqueness of positive square roots implies that A-1/2XA-1/2=A-1/2YA-1/2, i.e.,X=Y.

Theorem 0.14 (Maximum property of geometric mean)For A,B>0,


where the maximum is taken with respect to the positive semidefinite ordering.

If XA-1XB, then


and A-1/2XA-1/2(A-1/2BA-1/2)1/2i.e. XA#Bby Theorem .

Recall the fact that iff:[a,b]iscontinuousandAnAwithSp(An)[a,b]forallnN,thenSp(A)[a,b]andf(An)f(A).

Lemma 0.15Let A,B,C,D,An,BnB()++for all n.

  1. If ACand BD, then A#BC#D.

  2. If AnAand BnB, then An#BnA#B.

  3. If AnAand BnB, then limAn#Bnexists and does not depend on the choices of An,Bn.

(1) The extremal characterization allows us to prove only that (A#B)C-1(A#B)D. Indeed,


(2) Assume AnAand BnB. Then An#Bnis a decreasing sequence of strictly positive operators which is bounded below by 0. The order completeness of B()implies that this sequence converges strongly to a positive operator. Since An-1A-1, the Löwner-Heinz's inequality assures that An-1/2A-1/2and hence An-1/2A-1/2for all n. Note also that BnB1for all n. Recall that the multiplication is jointly continuous in the strong-operator topology if the first variable is bounded in norm. So, An-1/2BnAn-1/2converges strongly to A-1/2BA-1/2. It follows that


Since An1/2is norm-bounded by A1/2by Löwner-Heinz's inequality, we conclude that


The proof of (3) is just the same as the case of harmonic mean.

We define the geometric meanof A,B0by


Then A#B0for any A,B0.

Theorem 0.16The geometric mean enjoys the following properties

  1. monotonicity: A1A2,B1B2A1#B1A2#B2.

  2. continuity from above: AnA,BnBAn#BnA#B.

  3. fixed point property: A#A=A.

  4. self-duality: (A#B)-1=A-1#B-1.

  5. symmetry: A#B=B#A.

  6. congruence invariance: ΓC(A)#ΓC(B)=ΓC(A#B)for all invertible C.

(1) Use the formula () and Lemma (1).

(2) Follows from Lemma and the definition of the geometric mean.

(3) The unique positive solution to the equation XA-1X=Ais X=A.

(4) The unique positive solution to the equation X-1A-1X-1=Bis X-1=A#B. But this equstion is equivalent to XAX=B-1. So, A-1#B-1=X=(A#B)-1.

(5) The equation XA-1X=Bhas the same solution to the equation XB-1X=Aby taking inverse in both sides.

(6) We have


Then apply Theorem .

The congruence invariance asserts that ΓCis an isomorphism on B()++with respect to the operation of taking the geometric mean.

Lemma 0.17For A>0and B0, the operator


is positive if and only if B-C*A-1Cis positive, i.e., BC*A-1C.

By setting


we compute


Since ΓGpreserves positivity, we obtain the desired result.

Theorem 0.18The geometric mean A#Bof A,BB()+is the largest operator XB()safor which the operator


is positive.

By continuity argumeny, we may assume that A,B>0. If X=A#B, then the operator () is positive by Lemma . Let XB()sabe such that the operator () is positive. Then Lemma again implies that XA-1XBand


The Löwner-Heinz's inequality forces A-1/2XA-1/2(A-1/2BA-1/2)1/2. Now, applying ΓA1/2yields XA#B.

Remark 0.19The arithmetric mean and the harmonic mean can be easily defined for multivariable positive operators. The case of geometric mean is not easy, even for the case of matrices. Many authors tried to defined geometric means for multivariable positive semidefinite matrices but there is no satisfactory definition until 2004 in [20].

5. Operator connections

We see that the arithmetic, harmonic and geometric means share the properties (A1)–(A9) in common. A mean in general should have algebraic, order and topological properties. Kubo and Ando [10] proposed the following definition:

Definition 0.20A connectionon B()+is a binary operation σon B()+satisfying the following axioms for all A,A',B,B',CB()+:

  1. monotonicity: AA',BB'AσBA'σB'

  2. transformer inequality: C(AσB)C(CAC)σ(CBC)

  3. joint continuity from above: if An,BnB()+satisfy AnAand BnB, then AnσBnAσB.

The term “connection" comes from the study of electrical network connections.

Example 0.21The following are examples of connections:

  1. the left trivial mean(A,B)Aand the right trivial mean(A,B)B

  2. the sum (A,B)A+B

  3. the parallel sum

  4. arithmetic, geometric and harmonic means

  5. the weighed arithmetic meanwith weight α[0,1]which is defined for each A,B0by AαB=(1-α)A+αB

  6. the weighed harmonic meanwith weight α[0,1]which is defined for each A,B>0by A!αB=[(1-α)A-1+αB-1]-1and extended to the case A,B0by continuity.

From now on, assume dim=. Consider the following property:

  1. separate continuity from above: if An,BnB()+satisfy AnAand BnB, then AnσBAσBand AσBnAσB.

The condition (M3') is clearly weaker than (M3). The next theorem asserts that we can improve the definition of Kubo-Ando by replacing (M3) with (M3') and still get the same theory. This theorem also provides an easier way for checking a binary opertion to be a connection.

Theorem 0.22If a binary operation σon B()+satisfies (M1), (M2) and (M3'), then σsatisfies (M3), that is, σis a connection.

Denote by OM(+)the set of operator monotone functions from +to +. If a binary operation σhas a property (A), we write σBO(A). The following properties for a binary operation σand a function f:++play important roles:

  1. : If a projection PB()+commutes with A,BB()+, then

  2. : f(t)I=Iσ(tI)for any t+.

Proposition 0.23The transformer inequality (M2) implies

  1. Congruence invariance: For A,B0and C>0, C(AσB)C=(CAC)σ(CBC);

  2. Positive homogeneity: For A,B0and α(0,), α(AσB)=(αA)σ(αB).

For A,B0and C>0, we have


and hence (CAC)σ(CBC)C(AσB)C. The positive homogeneity comes from the congruence invariance by setting C=αI.

Lemma 0.24Let f:++be an increasing function. If σsatisfies the positive homogeneity, (M3') and (F), then fis continuous.

To show that fis right continuous at each t+, consider a sequence tnin +such that tnt. Then by (M3')


i.e. f(tn)f(t). To show that fis left continuous at each t>0, consider a sequence tn>0such that tnt. Then tn-1t-1and


Since fis increasing, tn-1f(tn)is decreasing. So, tt-1f(t)and fare left continuous.

Lemma 0.25Let σbe a binary operation on B()+satisfying (M3') and (P). If f:++is an increasing continuous function such that σand fsatisfy (F), then f(A)=IσAfor any AB()+.

First consider AB()+in the form i=1mλiPiwhere {Pi}i=1mis an orthogonal family of projections with sum Iand λi>0for all i=1,,m. Since each Picommutes with A, we have by the property (P) that


Now, consider AB()+. Then there is a sequence Anof strictly positive operators in the above form such that AnA. Then IσAnIσAand f(An)converges strongly to f(A). Hence, IσA=limIσAn=limf(An)=f(A).

Proof of Theorem:Let σBO(M1,M2,M3'). As in [10], the conditions (M1) and (M2) imply that σsatisfies (P) and there is a function f:++subject to (F). If 0t1t2, then by (M1)


i.e. f(t1)f(t2). The assumption (M3') is enough to guarantee that fis continuous by Lemma . Then Lemma results in f(A)=IσAfor all A0. Now, (M1) and the fact that dim=yield that fis operator monotone. If there is another gOM(+)satisfying (F), then f(t)I=IσtI=g(t)Ifor each t0, i.e. f=g. Thus, we establish a well-defined map σBO(M1,M2,M3')fOM(+)such that σand fsatisfy (F).

Now, given fOM(+), we construct σfrom the integral representation () in Proposition . Define a binary operation σ:B()+×B()+B()+by


where the integral is taken in the sense of Bochner. Consider A,BB()+and set Ft=A!tBfor each t[0,1]. Since AAIand BBI, we get


By Banach-Steinhaus' theorem, there is an M>0such that FtMfor all t[0,1]. Hence,


So, Ftis Bochner integrable. Since Ft0for all t[0,1], [0,1]Ftdμ(t)0. Thus, AσBis a well-defined element in B()+. The monotonicity (M1) and the transformer inequality (M2) come from passing the monotonicity and the transformer inequality of the weighed harmonic mean through the Bochner integral. To show (M3'), let AnAand BnB. Then An!tBA!tBfor t[0,1]by the monotonicity and the separate continuity from above of the weighed harmonic mean. Let ξH. Define a bounded linear mapΦ:B()by(T) =T,.ForeachnN,setTn(t) = An   !t   BandputT(t) = A   !t   B.ThenforeachnN{},TnisBochnerintegrableandSinceTn(t)T(t),wehavethatTn(t),T(t),asnforeacht[0,1].WeobtainfromthedominatedconvergencetheoremthatSo,An   BA   B.Similarly,A   BnA   B .Thus,satisfies(M3').Itiseasytoseethatf(t) I = I   (t I )fort0.Thisshowsthatthemapfissurjective.To show the injectivity of this map, let σ1,σ2BO(M1,M2,M3')be such that σifwhere, for each t0, Iσi(tI)=f(t)I,i=1,2.Since σisatisfies the property (P), we have IσiA=f(A)for A0by Lemma . Since σisatisfies the congruence invariance, we have that for A>0and B0,


For each A,B0, we obtain by (M3') that


where AϵA+ϵI. That is σ1=σ2. Therefore, there is a bijection between OM(+)and BO(M1,M2,M3'). Every element in BO(M1,M2,M3')admits an integral representation (). Since the weighed harmonic mean possesses the joint continuity (M3), so is any element in BO(M1,M2,M3').

The next theorem is a fundamental result of [10].

Theorem 0.26There is a one-to-one correspondence between connections σand operator monotone functions fon the non-negative reals satisfying


There is a one-to-one correspondence between connections σand finite Borel measures νon [0,]satisfying


Moreover, the map σfis an affine order-isomorphism between connections and non-negative operator monotone functions on +. Here, the order-isomorphism means that when σififor i=1,2, Aσ1BAσ2Bfor all A,BB()+if and only if f1f2.

Each connection σon B()+produces a unique scalar function on +, denoted by the same notation, satisfying


Let s,t+. If s>0, then sσt=sf(t/s). If t>0, then sσt=tf(s/t).

Theorem 0.27There is a one-to-one correspondence between connections and finite Borel measures on the unit interval. In fact, every connection takes the form


for some finite Borel measure μon [0,1]. Moreover, the map μσis affine and order-preserving. Here, the order-presering means that when μiσi(i=1,2), if μ1(E)μ2(E)for all Borel sets Ein [0,1], then Aσ1BAσ2Bfor all A,BB()+.

The proof of the first part is contained in the proof of Theorem . This map is affine because of the linearity of the map μfdμon the set of finite positive measures and the bijective correspondence between connections and Borel measures. It is straight forward to show that this map is order-preserving.

Remark 0.28Let us consider operator connections from electrical circuit viewpoint. A general connection represents a formulation of making a new impedance from two given impedances. The integral representation () shows that such a formulation can be described as a weighed series connection of (infinite) weighed harmonic means. From this point of view, the theory of operator connections can be regarded as a mathematical theory of electrical circuits.

Definition 0.29Let σbe a connection. The operator monotone function fin () is called the representing functionof σ. If μis the measure corresponds to σin Theorem , the measure μψ-1that takes a Borel set Ein [0,]to μ(ψ-1(E))is called the representing measureof σin the Kubo-Ando's theory. Here, ψ:[0,1][0,]is a homeomorphism tt/(1-t).

Since every connection σhas an integral representation (), properties of weighed harmonic means reflect properties of a general connection. Hence, every connection σsatisfies the following properties for all A,B0,TB()and invertible XB():

  1. transformer inequality: T*(AσB)T(T*AT)σ(T*BT);

  2. congruence invariance: X*(AσB)X=(X*AX)σ(X*BX);

  3. concavity: (tA+(1-t)B)σ(tA'+(1-t)B')t(AσA')+(1-t)(BσB')for t[0,1].

Moreover, if A,B>0,


and, in general, for each A,B0,


where AϵA+ϵIand BϵB+ϵI. These properties are useful tools for deriving operator inequalities involving connections. The formulas () and () give a way for computing the formula of connection from its representing function.

Example 0.30

  1. The left- and the right-trivial means have representing functions given by t1and tt, respectively. The representing measures of the left- and the right-trivial means are given respectively by δ0and δwhere δxis the Dirac measure at x. So, the α-weighed arithmetic mean has the representing function t(1-α)+αtand it has (1-α)δ0+αδas the representing measure.

  2. The geometric mean has the representing function tt1/2.

  3. The harmonic mean has the representing function t2t/(1+t)while tt/(1+t)corrsponds to the parallel sum.

Remark 0.31The map σμ, where μis the representing measure of σ, is not order-preserving in general. Indeed, the representing measure of is given by μ=(δ0+δ)/2while the representing measure of !is given by δ1. We have !butδ1μ.


6. Operator means

According to [1], a (scalar) mean is a binary operation Mon (0,)such that M(s,t)lies between sand tfor any s,t>0. For a connection, this property is equivalent to various properties in the next theorem.

Theorem 0.32The following are equivalent for a connection σon B()+:

  1. σsatisfies the betweenness property, i.e. ABAAσBB.

  2. σsatisfies the fixed point property, i.e. AσA=Afor all AB()+.

  3. σis normalized, i.e. IσI=I.

  4. the representing function fof σis normalized, i.e. f(1)=1.

  5. the representing measure μof σis normalized, i.e. μis a probability measure.

Clearly, (i) (iii) (iv). The implication (iii) (ii) follows from the congruence invariance and the continuity from above of σ. The monotonicity of σis used to prove (ii) (i). Since


we obtain that (iv) (v) (iii).

Definition 0.33A meanis a connection satisfying one, and thus all, of the properties in the previous theorem.

Hence, every mean in Kubo-Ando's sense satisfies the desired properties (A1)–(A9) in Section 3. As a consequence of Theorem , a convex combination of means is a mean.

Theorem 0.34Given a Hilbert space , there exist affine bijections between any pair of the following objects:

  1. the means on B()+,

  2. the operator monotone functions f:++such that f(1)=1,

  3. the probability Borel measures on [0,1].

Moreover, these correspondences between (i) and (ii) are order isomorphic. Hence, there exists an affine order isomorphism between the means on the positive operators acting on different Hilbert spaces.

Follow from Theorems and .

Example 0.35The left- and right-trivial means, weighed arithmetic means, the geometric mean and the harmonic mean are means. The parallel sum is not a mean since its representing function is not normalized.

Example 0.36The function ttαis an operator monotone function on +for each α[0,1]by the Löwner-Heinz's inequality. So it produces a mean, denoted by #α, on B()+. By the direct computation,


i.e. #αis the α-weighed geometric mean on +. So the α-weighed geometric mean on +is really a Kubo-Ando mean. The α-weighed geometric meanon B()+is defined to be the mean corresponding to that mean on +. Since tαhas an integral expression


(see [14]) where mdenotes the Lebesgue measure, the representing measure of #αis given by


Example 0.37Consider the operator monotone function


The direct computation shows that


which is the α-weighed harmonic mean. We define the α-weighed harmonic meanon B()+to be the mean corresponding to this operator monotone function.

Example 0.38Consider the operator monotone function f(t)=(t-1)/logtfor t>0,t1, f(0)0and f(1)1. Then it gives rise to a mean, denoted by λ, on B()+. By the direct computation,


i.e. λis the logarithmic mean on +. So the logarithmic mean on +is really a mean in Kubo-Ando's sense. The logarithmic meanon B()+is defined to be the mean corresponding to this operator monotone function.

Example 0.39The map t(tr+t1-r)/2is operator monotone for any r[0,1]. This function produces a mean on B()+. The computation shows that


However, the corresponding mean on B()+is not given by the formula


since it is not a binary operation on B()+. In fact, the formula () is considered in [21], called the Heinz meanof Aand B.

Example 0.40For each p[-1,1]and α[0,1], the map


is an operator monotone function on +. Here, the case p=0is understood that we take limit as p0. Then


The corresponding mean on B()+is called the quasi-arithmetic power meanwith parameter (p,α), defined for A>0and B0by


The class of quasi-arithmetic power means contain many kinds of means: The mean #1,αis the α-weighed arithmetic mean. The case #0,αis the α-weighed geometric mean. The case #-1,αis the α-weighed harmonic mean. The mean #p,1/2is the power meanor binomial meanof order p. These means satisfy the property that


Moreover, they are interpolated in the sense that for all p,q,α[0,1],


Example 0.41If σ1,σ2are means such that σ1σ2, then there is a family of means that interpolates between σ1and σ2, namely, (1-α)σ1+ασ2for all α[0,1]. Note that the map α(1-α)σ1+ασ2is increasing. For instance, the Heron meanwith weight α[0,1]is defined to be hα=(1-α)#+α. This family is the linear interpolations between the geometric mean and the arithmetic mean. The representing function of hαis given by


The case α=2/3is called the Heronian meanin the literature.

7. Applications to operator monotonicity and concavity

In this section, we generalize the matrix and operator monotonicity and concavity in the literature (see e.g. [3], [22]) in such a way that the geometric mean, the harmonic mean or specific operator means are replaced by general connections. Recall the following terminology. A continuous function f:Iis called an operator concave functionif


for any t[0,1]and Hermitian operators A,BB()whose spectrums are contained in the interval Iand for all Hilbert spaces . A well-known result is that a continuous function f:++is operator monotone if and only if it is operator concave. Hence, the maps ttrand tlogtare operator concave for r[0,1]. Let and 𝒦be Hilbert spaces. A map Φ:B()B(𝒦)is said to be positiveif Φ(A)0whenever A0. It is called unitalif Φ(I)=I. We say that a positive map Φis strictly positiveif Φ(A)>0when A>0. A map Ψfrom a convex subset 𝒞of B()sato B(𝒦)sais called concaveif for each A,B𝒞and t[0,1],


A map Ψ:B()saB(𝒦)sais called monotoneif ABassures Ψ(A)Ψ(B). So, in particular, the map AAris monotone and concave on B()+for each r[0,1]. The map AlogAis monotone and concave on B()++.

Note first that, from the previous section, the quasi-arithmetic power mean (A,B)A#p,αBis monotone and concave for any p[-1,1]and α[0,1]. In particular, the following are monotone and concave:

  1. any weighed arithmetic mean,

  2. any weighed geometric mean,

  3. any weighed harmonic mean,

  4. the logarithmic mean,

  5. any weighed power mean of order p[-1,1].

Recall the following lemma from [22].

Lemma 0.42 (Choi's inequality)If Φ:B()B(𝒦)is linear, strictly positive and unital, then for every A>0, Φ(A)-1Φ(A-1).

Proposition 0.43If Φ:B()B(𝒦)is linear and strictly positive, then for any A,B>0


For each XB(), set Ψ(X)=Φ(A)-1/2Φ(A1/2XA1/2)Φ(A)-1/2.Then Ψis a unital strictly positive linear map. So, by Choi's inequality, Ψ(A)-1Ψ(A-1)for all A>0. For each A,B>0, we have by Lemma that


So, we have the claim.

Theorem 0.44If Φ:B()B(𝒦)is a positive linear map which is norm-continuous, then for any connection σon B(𝒦)+and for each A,B>0,


If, addition, Φis strongly continuous, then () holds for any A,B0.

First, consider A,B>0. Assume that Φis strictly positive. For each XB(), set


Then Ψis a unital strictly positive linear map. So, by Choi's inequality, Ψ(C)-1Ψ(C-1)for all C>0. For each t[0,1], put Xt=B-1/2(A!tB)B-1/2>0. We obtain from the previous proposition that


For general case of Φ, consider the family Φϵ(A)=Φ(A)+ϵIwhere ϵ>0. Since the map (A,B)A!tB=[(1-t)A-1+tB-1]-1is norm-continuous, we arrive at


For each connection σ, since Φis a bounded linear operator, we have


Suppose further that Φis strongly continuous. Then, for each A,B0,


The proof is complete.

As a special case, ifΦ:Mn(Mn(is a positive linear map, then for any connection σand for any positive semidefinite matricesA,BMn(, we have


In particular, Φ(A)#p,αΦ(B)Φ(A)#p,αΦ(B)for any p[-1,1]and α[0,1].

Theorem 0.45If Φ1,Φ2:B()+B(𝒦)+are concave, then the map


is concave for any connection σon B(𝒦)+.

Let A1,A1',A2,A2'0and t[0,1]. The concavity of Φ1and Φ2means that for i=1,2


It follows from the monotonicity and concavity of σthat


This shows the concavity of the map (A1,A2)Φ1(A1)σΦ2(A2).

In particular, if Φ1and Φ2are concave, then so is (A,B)Φ1(A)#p,αΦ2(B)for p[-1,1]and α[0,1].

Corollary 0.46Let σbe a connection. Then, for any operator monotone functions f,g:++, the map (A,B)f(A)σg(B)is concave. In particular,

  1. the map (A,B)ArσBsis concave on B()+for any r,s[0,1],

  2. the map (A,B)(logA)σ(logB)is concave on B()++.

Theorem 0.47If Φ1,Φ2:B()+B(𝒦)+are monotone, then the map


is monotone for any connection σon B(𝒦)+.

Let A1A1'and A2A2'. Then Φ1(A1)Φ1(A1')and Φ2(A2)Φ2(A2')by the monotonicity of Φ1and Φ2. Now, the monotonicity of σforces Φ1(A1)σΦ2(A2)Φ1(A1')σΦ2(A2').

In particular, if Φ1and Φ2are monotone, then so is (A,B)Φ1(A)#p,αΦ2(B)for p[-1,1]and α[0,1].

Corollary 0.48Let σbe a connection. Then, for any operator monotone functions f,g:++, the map (A,B)f(A)σg(B)is monotone. In particular,

  1. the map (A,B)ArσBsis monotone on B()+for any r,s[0,1],

  2. the map (A,B)(logA)σ(logB)is monotone on B()++.

Corollary 0.49Let σbe a connection on B()+. If Φ1,Φ2:B()+B()+is monotone and strongly continuous, then the map


is a connection on B()+. Hence, the map


is a connection for any operator monotone functions f,g:++.

The monotonicity of this map follows from the previous result. It is easy to see that this map satisfies the transformer inequality. Since Φ1and Φ2strongly continuous, this binary operation satisfies the (separate or joint) continuity from above. The last statement follows from the fact that if AnA, then Sp(An)[0,A1]for all nand hence f(An)f(A).

8. Applications to operator inequalities

In this section, we apply Kubo-Ando's theory in order to get simple proofs of many classical inequalities in the context of operators.

Theorem 0.50 (AM-LM-GM-HM inequalities)For A,B0, we have


It is easy to see that, for each t>0,t1,


Now, we apply the order isomorphism which converts inequalities of operator monotone functions to inequalities of the associated operator connections.

Theorem 0.51 (Weighed AM-GM-HM inequalities)For A,B0and α[0,1], we have


Apply the order isomorphism to the following inequalities:


The next two theorems are given in [23].

Theorem 0.52For each i=1,,n, let Ai,BiB()+. Then for each connection σ


Use the concavity of σtogether with the induction.

By replacing σwith appropriate connections, we get some interesting inequalities.

(1) Cauchy-Schwarz's inequality: For Ai,BiB()sa,


(2) Hölder's inequality: For Ai,BiB()+and p,q>0such that 1/p+1/q=1,


(3) Minkowski's inequality: For Ai,BiB()++,


Theorem 0.53Let Ai,BiB()+, i=1,,n, be such that




Substitute A1to A1-A2--Anand B1to B1-B2--Bnin ().

Here are consequences.

(1) Aczél's inequality: For Ai,BiB()sa, if




(2) Popoviciu's inequality: For Ai,BiB()+and p,q>0such that 1/p+1/q=1, if p,q>0are such that 1/p+1/q=1and




(3) Bellman's inequality: For Ai,BiB()++, if




The mean-theoretic approach can be used to prove the famous Furuta's inequality as follows. We cite [24] for the proof.

Theorem 0.54 (Furuta's inequality)For AB0, we have


where r0,p0,q1and (1+2r)qp+2r.

By the continuity argument, assume that A,B>0. Note that () and () are equivalent. Indeed, if () holds, then () comes from applying () to A-1B-1and taking inverse on both sides. To prove (), first consider the case 0p1. We have Bp+2r=BrBpBrBrApBrand the Löwner-Heinz's inequality (LH) implies the desired result. Now, consider the case p1and q=(p+2r)/(1+2r), since () for q>(p+2r)/(1+2r)can be obtained by (LH). Let f(t)=t1/qand let σbe the associated connection (in fact, σ=#1/q). Must show that, for any r0,


For 0r12, we have by (LH) that A2rB2rand


Now, set s=2r+12and q1=(p+2s)/(1+2s)1. Let f1(t)=t1/q1and consider the associated connection σ1. The previous step, the monotonicity and the congruence invariance of connections imply that


Note that the above result holds for A,B0via the continuity of a connection. The desired equation () holds for all r0by repeating this process.


The author thanks referees for article processing.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Pattrawut Chansangiam (July 11th 2012). Operator Means and Applications, Linear Algebra - Theorems and Applications, Hassan Abid Yasser, IntechOpen, DOI: 10.5772/46479. Available from:

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