Obstructions to the uniform stability of a C 0-semigroup

Abstract

Let T t : X → X be a C 0 -semigroup with generator A. We prove that if the abscissa of uniform boundedness of the resolvent s 0 (A) is greater than zero then for each nondecreasing function h(s) :then such x may be taken in D(A ∞ ).

On weak decay rates and uniform stability of bounded linear operators

“…We should point out that, at least in the case E = X ′ , Theorem 5.1 can be obtained as a corollary from a recent result of Storozhuk on the size of weak semigroup orbits [12,Theorem 1]. However, since the proof of Storozhuk's result is technically rather involved, we prefer to give a more direct argument here.…”

“…A very recent contribution to the field is [1] where uniform stability of a C 0 -semigroup is derived from certain discrete, weak summability conditions. Some rather general results on the existence of large weak orbits of C 0 -semigroups were recently published in [12] and [7] While van Neerven used Banach function spaces and in particular Orlicz spaces to prove his result quoted at the beginning, we use a different approach: In Section 3, we show that an operator T satisfies r(T ) < 1 if all sequences ( x ′ , T n x ) n∈N0 (x ∈ X, x ′ ∈ X ′ ) are in some appropriate manner controlled by a sequence f which converges to 0. In Proposition 3.4 we show that the situation does not change if f is allowed to vary within a countable set of sequences.…”

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…This is in accordance with Theorem 2.5 stated above. For more results on lower bounds for weak orbits of C 0 -semigroups on mostly reflexive Banach spaces, see [80,81].…”

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…This is in accordance with Theorem 2.5 stated above. For more results on lower bounds for weak orbits of C 0 -semigroups on mostly reflexive Banach spaces, see [114] and [141].…”