scholarly journals Lyapunov operator inequalities for exponential stability of Banach space semigroups of operators

2012 ◽  
Vol 25 (3) ◽  
pp. 401-403
Author(s):  
Ciprian Preda ◽  
Petre Preda
Keyword(s):  
Banach Space ◽  
Exponential Stability ◽  
Semigroups Of Operators ◽  
Operator Inequalities ◽  
Lyapunov Operator

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 Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Aftab Khan ◽  
Gul Rahmat ◽  
Akbar Zada
Keyword(s):  
Banach Space ◽  
Exponential Stability ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Uniform Exponential Stability ◽  
Discrete Semigroup ◽  
Exponentially Stable ◽  
Discrete Semigroups

We prove that a discrete semigroup𝕋={T(n):n∈ℤ+}of bounded linear operators acting on a complex Banach spaceXis uniformly exponentially stable if and only if, for eachx∈AP0(ℤ+,X), the sequencen↦∑k=0n‍T(n-k)x(k):ℤ+→Xbelongs toAP0(ℤ+,X). Similar results for periodic discrete evolution families are also stated.

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 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.

scholarly journals Functional differential equations with infinite delay in Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 497-508 ◽  
Author(s):  
Jin Liang ◽  
Tijun Xiao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Exponential Stability ◽  
Functional Differential Equation ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Zero Solution ◽  
Fading Memory ◽  
Fundamental Operator ◽  
Functional Differential

In this paper, a definition of the fundamental operator for the linear autonomous functional differential equation with infinite delay in a Banach space is given, and some sufficient and necessary conditions of the fundamental operator being exponentially stable in abstract phase spaces which satisfy some suitable hypotheses are obtained. Moreover, we discuss the relation between the exponential asymptotic stability of the zero solution of nonlinear functional differential equation with infinite delay in a Banach space and the exponential stability of the solution semigroup of the corresponding linear equation, and find that the exponential stability problem of the zero solution for the nonlinear equation can be discussed only in the exponentially fading memory phase space.

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

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