Semigroups of operators and mixed properties of Banach space elements

1974 ◽  
Vol 16 (1) ◽  
pp. 649-654
Author(s):  
A. P. Terekhin
Keyword(s):  
Banach Space ◽  
Semigroups Of Operators

scholarly journals Ergodicity and differences of functions on semigroups

1998 ◽  
Vol 64 (2) ◽  
pp. 253-265 ◽  
Author(s):  
Bolis Basit ◽  
A. J. Pryde
Keyword(s):  
Banach Space ◽  
General Framework ◽  
Closed Subspace ◽  
Semigroups Of Operators ◽  
Locally Compact Abelian Group ◽  
General Notion ◽  
Abelian Semigroup ◽  
Locally Compact ◽  
Translation Invariant ◽  
Bounded Functions

AbstractIseki [11] defined a general notion of ergodicity suitable for functions ϕ: J → X where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established by Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let A be a translation invariant closed subspace of the space of all bounded functions from J to X. We prove that if A contains the constant functions and ϕ is an ergodic function whose differences lie in A then ϕ ∈ A. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of ergodic vectors of a representation T: J → L(X) generalizing results known for the case J = Z+. Finally, when J is a subsemigroup of a locally compact abelian group G, we compare the Iseki integrals with the better known Cesàro integrals.

scholarly journals Operator calculus on the class of Sato’s hyperfunctions

2013 ◽  
Vol 5 (1) ◽  
pp. 114-120
Author(s):  
M.I. Patra ◽  
S.V. Sharyn
Keyword(s):  
Banach Space ◽  
Compact Support ◽  
Functional Calculus ◽  
Analytic Semigroups ◽  
Semigroups Of Operators ◽  
Operator Calculus ◽  
Symbol Class

We construct a functional calculus for generators of analytic semigroups of operators on a Banach space. The symbol class of the calculus consists of hyperfunctions with a compact support in $[0,\infty)$. Domain of constructed calculus is dense in the Banach space.

scholarly journals Exponential dichotomy of strongly discontinuous semigroups

1984 ◽  
Vol 30 (3) ◽  
pp. 435-448 ◽  
Author(s):  
P. Preda ◽  
M. Megan
Keyword(s):  
Banach Space ◽  
Exponential Stability ◽  
General Class ◽  
Exponential Dichotomy ◽  
Continuous Semigroup ◽  
Sufficient Conditions ◽  
Semigroups Of Operators ◽  
Strongly Continuous Semigroup ◽  
Strongly Continuous Semigroups ◽  
Necessary And Sufficient

In this paper we give necessary and sufficient conditions for exponential dichotomy of a general class of strongly continuous semigroups of operators defined on a Banach space. As a particular case we obtain a Datko theorem for exponential stability of a strongly continuous semigroup of class C0 defined on a Banach space.

scholarly journals On a Certain Class of Semigroups of Operators

2011 ◽  
Vol 18 (02) ◽  
pp. 129-142 ◽  
Author(s):  
Paolo Aniello
Keyword(s):  
Banach Space ◽  
Compact Group ◽  
Banach Spaces ◽  
Locally Compact Group ◽  
Semigroups Of Operators ◽  
Probability Measures ◽  
Locally Compact ◽  
Quantum Dynamical ◽  
Interesting Class ◽  
Natural Way

We define an interesting class of semigroups of operators in Banach spaces, namely, the randomly generated semigroups. This class contains as a remarkable subclass a special type of quantum dynamical semigroups introduced in the early 1970s by Kossakowski. Each randomly generated semigroup is associated, in a natural way, with a pair formed by a representation or an antirepresentation of a locally compact group in a Banach space and by a convolution semigroup of probability measures on this group. Examples of randomly generated semigroups having important applications in physics are briefly illustrated.

Some ergodic theorems for one-parameter semigroups of operators

1956 ◽  
Vol 249 (962) ◽  
pp. 151-177 ◽  
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Strong Convergence ◽  
Resolvent Operator ◽  
Infinitesimal Generator ◽  
Semigroups Of Operators ◽  
Ergodic Theorems ◽  
Semigroup Of Operators ◽  
Ergodic Properties ◽  
The Resolvent Operator

If ^ = {7^: t ^0} is a one-parameter semigroup of operators on a Banach space X , an element x of X is called ergodic if T t X has a generalized limit as t -> oo. It is shown, for a wide class of semigroups, that the use of Abel or Cesaro limits, and of weak or strong convergence, leads to four equivalent definitions of ergodicity. When the resolvent operator of G has suitable compactness properties, every element of X is ergodic. The ergodic properties of G can be completely determined when its infinitesimal generator is known. Some of these results can be extended to more generaltypes of weak convergence in X , and this leads to a discussion of ergodic properties of the semigroup adjoint to G

scholarly journals Lifting unconditionally converging series and semigroups of operators

1998 ◽  
Vol 57 (1) ◽  
pp. 135-146 ◽  
Author(s):  
Manuel González ◽  
Antonio Martínez-Abejón
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Equivalent Norm ◽  
Semigroups Of Operators ◽  
Perturbation Class

We introduce and study two semigroups of operators u+ and u_, defined in terms of unconditionally converging series. We prove a lifting result for unconditionally converging series that allows us to show examples of operators in u+. We obtain perturbative characterisations for these semigroups and, as a consequence, we derive characterisations for some classes of Banach spaces in terms of the semigroups. If u+(X, Y) is non-empty and every copy of c0 in Y is complemented, then the same is true in X. We solve the perturbation class problem for the semigroup u_, and we show that a Banach space X contains no copies of ℓ∞ if and only if for every equivalent norm |·| on X, the semiembeddings of (X, |·|) belong to u+.

Monotone Semigroups of Operators on Cones*

1969 ◽  
Vol 12 (3) ◽  
pp. 299-309
Author(s):  
David W. Boyd
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Special Class ◽  
Continuous Mapping ◽  
Natural Generalization ◽  
Linear Operators ◽  
Sequence Spaces ◽  
Semigroups Of Operators ◽  
Interpolation Spaces

In this paper we consider a special class of linear operators defined on a cone K in a Banach space X. This class of operators is the natural generalization of a class of operators which has applications in the theory of interpolation spaces. In particular, using the criteria developed in Theorem 1, it is possible to characterize those sequence spaces X such that every linear operator A of weak types (p, p) and (q, q) is a continuous mapping of X into itself. For details of this we refer the reader to [3].

Semigroups of operators and an application to spectral theory

1984 ◽  
Vol 96 (1) ◽  
pp. 143-149 ◽  
Author(s):  
W. Ricker
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Spectral Theory ◽  
Continuous Linear Operator ◽  
Integral Transforms ◽  
Fundamental Importance ◽  
Semigroups Of Operators ◽  
Continuous Linear ◽  
Scalar Type

A problem of fundamental importance in Spectral Theory consists of finding criteria for an operator to be of scalar-type in the sense of N. Dunford [1]. One relatively general approach in determining such criteria is based on the method of integral transforms (see for example [4], [5], [6], [11], [12]). For example, if X is a Banach space and T is a continuous linear operator on X, then the group {eitT; t real} exists. As noted by several authors (e.g. [4], [6]), this group can then be effectively used for analysing the operator T.

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