On the Decay of Bounded Semigroup on the Domain of its Generator

Abstract

For a bounded C 0 -semigroup on a Banach space X, we prove the following statement: the rate of decay of the semigroup on the domain of its generator is bounded by some decreasing function if and only if the spectrum of the semigroup does not contain any pure imaginary points. Our approach is based on the analysis of a special semigroup on the space of bounded linear operators L (X, X).

On the extension of battys theorem on the semigroup asymptotic stability

“…a constructive proof of existence of such a function f satisfying (4) and ( 5) for an arbitrary semigroup is given in [10]; without loss of generality we only prove the Theorem (3) in the case of ω 0 = 0. Indeed, for arbitrary ω 0 one can consider the shifted semigroup {e −ω 0 t T (t)} t≥0 ; we will clarify the connection between (7) and (2) at the end of the proof.…”

“…In the proof we will use the construction of the special operator-valued semigroup introduced in [10]. Let X ⊂ L (X) be defined as…”

“…Important properties of this semigroup were shown in [10], namely that { T (t)} t≥0 forms a C 0 -semigroup on X, and that for A and A being the generators of {T (t)} t≥0 and { T (t)} t≥0 , respectively, it holds that…”

“…Moreover, the semi-uniform stability may occur even for unbounded semigroups (see [9], for example). For the case of unbounded semigroups it was shown in [10] that the condition (1) remains necessary for T (t)A −1 → 0. On the other hand for an unbounded semigroup T with ω 0 (T ) ≥ 0 the concept of semiuniform stability led us to consider a more general property:…”

On Asymptotic Estimation of a Discrete Type $$C_0$$ C 0 -Semigroups on Dense Sets: Application to Neutral Type Systems

“…In particular, Batty [3,4] and Phong [10] (cf. Sklyar [15]) proved that for bounded C 0 -semigroup T (t) on a Banach space with the generator A, the following holds: if…”