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Operator Topology for Logarithmic Infinitesimal Generators

By Yoritaka Iwata

Submitted: October 21st 2019Reviewed: March 23rd 2020Published: May 11th 2020

DOI: 10.5772/intechopen.92226

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Abstract

Generally unbounded infinitesimal generators are studied in the context of operator topology. Beginning with the definition of seminorm, the concept of locally convex topological vector space is introduced as well as the concept of Fréchet space. These are the basis for defining operator topologies. Consequently, by associating the topological properties with the convergence of sequence, a suitable mathematical framework for obtaining the logarithmic representation of infinitesimal generators is presented.

Keywords

  • operator theory
  • locally strong topology
  • infinitesimal generator

1. Introduction

Let Xbe an infinite/finite dimensional Banach space with the norm , and Ybe a dense subspace of X. The Cauchy problem for abstract evolution equation of hyperbolic type [1, 2] is defined by

dut/dtAtut=ft,t0T,u0=u0E1

in X, where At: YXis assumed to be the infinitesimal generator of evolution operator Utssatisfying the strong continuity (for the definition of strong topology, refer to the following section) and the semigroup property:

Uts=UtrUrsE2

for 0srt<T. Utsis a two-parameter C0-semigroup of operator that is a generalization of one-parameter C0-semigroup and therefore an abstract generalization of the exponential function of operator. For an infinitesimal generator Atof Uts, the solution utis represented by ut=Utsuswith usXfor a certain 0sT(cf. Hille-Yosida Theorem; for example see [3, 4, 5]).

2. Operator topology

2.1 The dual formalism of evolution equation

The dual space of Xbeing denoted by Xis defined by

X=LXK,E3

where Kis a scalar field making up the space X, and LXKdenotes the space of continuous linear functionals. Since Kis also a Banach space, LXKsatisfies the properties of Banach space.

Let : X×Xbe a dual product between Xand X, and ℂ be a set of complex numbers. The adjoint operator At: DAtXis defined by the operator satisfying

Atuv=uAtvE4

for any uDAtand vDAt. If Xis a Hilbert space, the dual product is replaced with a scalar product equipped with X. Unique dual correspondence is valid, if Xis strictly convex Banach space at least (for convex Banach space, see [6]). By taking the dual product, the abstract evolution equation in X:

dut/dtAtut=ft,t0TE5

is written as a scalar-valued evolution equation in :

dut/dtvtAtutvt=ftvt,t0TE6

for a certain vtX. The formalism (6), which is associated with the Gelfand triplets [7], has been considered by variational method of abstract evolution equations [8, 9]. Eqs. (5) and (6) cannot necessarily be equivalent in the sense of operator topology.

2.2 Locally strong topology

Let pube a seminorm equipped with a space X, and the family of seminorms be denoted by P. Locally convex spaces are the generalization of normed spaces. Here the topology is called locally convex, if the topology admits a local base at 0 consisting of balanced, absorbent, and convex sets. In other words, a topological space Xis called locally convex, if its topology is generated by a family of seminorms satisfying

pPuXpu=0=0X,E7

where 0Xdenotes the zero of topological space X. Fréchet spaces are locally convex spaces that are completely metrizable with a certain complete metric. It follows that a Banach space Xis trivially a Fréchet space.

For Banach spaces Xand Y, the bounded linear operators from Xto Yis denoted by BXY. In particular BXXis written by BX. The operator space BXis called the Banach algebra, since it holds the structure of algebraic ring. The norm of BX, which means the operator norm, is defined by

TBX=supx0TxXxX.E8

A norm is trivially a seminorm. Consequently, the topological space in this article is set to be a Banach space BX.

There are several standard typologies defined on BX. The topologies listed below are all locally convex, which implies that they are defined by a family of seminorms. The topologies are identified by the convergence arguments. Let Tnbe a sequence in a Banach space X.

  • TnTin the uniform topology, if TnTBX0;

  • TnTin the strong topology, if TnxTxfor any xX;

  • TnTin the weak topology, if FTnxFTxfor any FXand xX;

where the uniform topology is the strongest, and the weak topology is the weakest. Indeed a topology is called stronger if it has more open sets and weaker if it has less open sets. If Yis a vector space of linear maps on the vector space X, then a topology σXYis defined to be the weakest topology on Xsuch that all elements of Yare continuous. The topology of σXYtype is apparent if the formalism (6) is considered; the weak topology is written by σBXBX. Although there are some intermediate topologies between the above three; strong* topology, weak* topology, and so on, another type of topology is newly introduced in this article.

Definition 1 (locally strong topology)

  • TnTin the locally strong topology, if Tnx¯Tx¯for a certain x¯X.

This topology is utilized to define a weak differential appearing in the logarithmic representation of infinitesimal generators.

3. Infinitesimal generator

3.1. Logarithmic infinitesimal generator

The logarithm of evolution operator is represented using the Riesz-Dunford integral. A time interval 0Twith 0s,tTis provided. For a certain usX, let a trajectory ut=Utsusbe given in a Banach space X. For a given UtsBX, its logarithm is well defined [10]; there exists a certain complex number κsatisfying

LogUts+κI=12πiΓLogλλκUts1,E9

where an integral path Γ, which excludes the origin, is a circle in the resolvent set of Uts+κI.

Let us call LogUts+κIthe alternative infinitesimal generator to At. Since the alternative infinitesimal generator [11]

atsLogUts+κIE10

is necessarily bounded on X, its exponential function eatsis always well defined as a convergent power series. Note that the alternative infinitesimal generator atsis bounded on X, although the corresponding infinitesimal generator Atis possibly an unbounded operator. It follows that eats=eats1is automatically well defined if eatsis well defined. Also eatsis invertible regardless of the validity of the invertible property for original Uts. The logarithmic representation of infinitesimal generator (logarithmic infinitesimal generator, for short) is obtained as follows.

Lemma 1 (Logarithmic infinitesimal generators [10]). Let tand ssatisfy 0t,sT, and Ybe a dense subspace of X. If Atand Utscommute, infinitesimal generators At0tTare represented by means of the logarithm function; there exists a certain complex number κ0such that

Atu=Iκeats1tatsu,E11

where uis an element of a dense subspace Yof X, and tis a kind weak differential being defined by the locally strong topology.

Proof. Only formal discussion is made here (for the detail, see [10, 12]).

Uts+κItats=Uts+κIUts+κI1tUts.E12

It leads to

Atu=Uts1tUtsu=Uts1Uts+κItatsu=I+κeatsκI1tatsu=eatsκI+κIeatsκI1tatsu=Iκeats1tatsu,E13

where uis an element in Y.

The solution trajectory is given, and {A(t)}0 ≤ t ≤ T are determined for one fixed u ∈ Y. Here is a reason why the locally-strong topology is introduced. Eq. (11) is the logarithmic representation of infinitesimal generator At. This representation is useful not only to mathematical analysis but also to operator algebra [12, 13].

3.2. Differential operator in the logarithmic representation

The convergence of the limit in the differential operator tof Eq. (11) is discussed. The convergence in the locally strong topology is applied to the evolution operator UtsBX.

Definition 2 (Weak limit using the locally strong topology). For 0t,sT, the weak limit

limh0h1Ut+hsUtsus=limh0h1Ut+htIUtsusE14

is assumed to exist for a certain usin a dense subspace Yof X. The limit “lim” is practically denoted by “wlim” in the following, since it is a limit in a kind of weak topology.

Let t-differential of Utsin a weak sense of the above be denoted by t, then it follows that

tUtsus=AtUtsus,E15

and a generalized concept of infinitesimal generator At:YXis introduced by

Atuswlimh0h1Ut+htIusE16

for a certain usY, where the convergence in wlimmust be replaced with the strong convergence in the standard theory of abstract evolution equations [4].

The operator Atdefined in this way for a whole family Uts0t,sTis called the pre-infinitesimal generator in [10], because only its exponetiability with a certain ideal domain is ensured without justifying the dense property of its domain space. Indeed pre-infinitesimal generators are not necessarily infinitesimal generators, while infinitesimal generators are pre-infinitesimal generators.

4. Main result

According to the standard theory of abstract evolution equation [4], the evolution operator is assumed to be strongly continuous. It follows that the trajectory Utsusis continuous in X. Here is the reason why it is sufficient to consider the convergence of differential operator tonly with a fixed element u¯=usYXwith 0sT. Also, in terms of analyzing the trajectory in finite-/infinite-dimensional dynamical systems, it is reasonable to consider the convergence in the topology uniquely sticking to the trajectory. Consequently the infinitesimal generator can be extracted by one sample point in the interval (Figure 1). Indeed, according to the independence between tand s,

Figure 1.

Trajectory Utsus in X. Uts∈BX is assumed to be strongly continuous with respect to time variables, so that a trajectory Utsus is continuous in X. Note that it is necessary to replace the trajectory Utsus with the regularized trajectory eats−κIus to consider the negative time evolution [11, 12].

Atus=wlimh0h1Ut+htIusE17

is true for any tsT, once Atis obtained for a sample point usY. Such a restrictive topological treatment contributes to generalize or weaken the differential.

For a given evolution operator UtsBX, the profile of locally strong topology is obtained in this article. In Banach space BX, a subset FBXis a closed set, if and only if

anF,aBX,anaaFE18

is satisfied n=12, where the operation of limit depends on a chosen topology. Here the following two theorems are proved to clarify the mathematical property of the locally strong topology.

Theorem 1.1. The locally strong topology is weaker than the strong topology.

Proof. It is enough to prove that a closed set in strong topology is closed in the locally strong topology. Let an arbitrary closed set of BXin the strong topology be V. It satisfies

TnV,TBX,limnTnxTx=0TVE19

for an arbitrary xX. In the locally strong topology (TnTx¯for a certain x¯X), the convergence TnTVis true.

Theorem 1.2. The locally strong topology is not necessarily stronger than the weak topology.

Proof. The proof is carried out in the similar manner to Theorem 1.1. Let an arbitrary closed set of BXin the locally strong topology be V. It satisfies

TnV,TBX,limnTnx¯Tx¯=0TVE20

for a fixed x¯X. By taking the dual product of an arbitrary FX, it follows that

TnV,aBX,limnTnTx¯F=0TVE21

It shows that the closedness of Vin a locally weak topology, where the locally weak topology is defined by fixing the weak topology with x=x¯in the same manner as the locally strong topology.

On the other hand, weak convergence cannot be assured if xx¯. Indeed, for x1x¯, the statement

TnV,aBX,limnTnTx1F=0TVE22

does not follow from the statement (20). It shows that there is no guarantee for locally strong topology to be stronger than the weak topology.

5. Summary

The concept of locally strong topology is introduced by the proofs clarifying its specific topological weakness. The locally strong topology is a topology unique to the solution trajectory of abstract evolution equations (Figure 1). That is, the locally strong topology holds the one-dimensionality specific to a certain trajectory. Although the locally strong topology has already been utilized even without the nomenclature to clarify the algebraic structure of semigroups of operators and their infinitesimal generators, those fundamentals are made in this article. The locally strong topology is also expected to be useful to analyze each single trajectory defined in finite-/infinite-dimensional dynamical systems.

Acknowledgments

The author is grateful to Prof. Emeritus Hiroki Tanabe for valuable comments. This work was partially supported by JSPS KAKENHI Grant No. 17K05440.

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Yoritaka Iwata (May 11th 2020). Operator Topology for Logarithmic Infinitesimal Generators [Online First], IntechOpen, DOI: 10.5772/intechopen.92226. Available from:

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