Obstructions for uniform stability of C_0-semigroup

TL;DR

This paper investigates conditions under which a C_0-semigroup's stability is obstructed, showing that certain spectral properties of its generator imply the existence of vectors with unbounded integral behavior.

Contribution

It establishes a link between the spectral properties of the generator's resolvent and the obstructions to uniform stability of the semigroup.

Findings

Non-negative abscissa of uniform boundedness implies unbounded integrals for certain vectors.

Presence of purely imaginary spectrum allows vectors in the domain of all powers of A.

Spectral conditions serve as obstructions to the semigroup's uniform stability.

Abstract

Let T be a C_0-semigroup on X with generator A. We prove that if the abscissa of uniform boundedness of the resolvent A is non-negative then for each a non-decreasing function h:[0,\infty] -> [0,\infty], there are x' in X' and x in X such that integral from 0 to \infty of h(|< x',T(t)x)>| is equal to \infty. If Sp(A) contained a number from iR then such x may be taken in D(A^\infty).