Approximation and uniform polynomial stability ofC₀-semigroups

Stability Conditions for Perturbed Semigroups on a Hilbert Space via Commutators

“…The literature on the stability of semigroups is very rich. The classical results are presented in the books [1,2], about the recent investigations for instance see [3] - [6], [7,8,9,10]. In particular, in [7] the author investigates the uniform, strong, weak and almost weak stabilities of multiplication semigroups on Banach space valued L p -spaces.…”

Direct integrals of strongly continuous operator semigroups

“…We motivate this final section by beginning with and stating a result concerning the stability of a uniformly bounded countable sequence of C 0semigroups on Hilbert spaces from [21] with an added assumption that we believe is necessary. Then for a fixed α > 0 the following conditions are equivalent:…”

“…The assumption (7.1) is missing from the statement of [21, Theorem 3.2] as found in [21]. It is ambiguous as to whether or not the authors of that paper implicitly assume this condition in [21,Theorem 3.2] without explicitly stating it.…”

“…The assumption (7.1) is missing from the statement of [21, Theorem 3.2] as found in [21]. It is ambiguous as to whether or not the authors of that paper implicitly assume this condition in [21,Theorem 3.2] without explicitly stating it. The introduction, preparatory work in Section 2, and application in Theorem 4.4 of their paper indicate that they are, in fact, assuming (7.1).…”

“…Sections 2-3 provide the preliminaries and basic operator-theoretic results needed to tackle the central issue, which, as mentioned, is addressed in Section 4. Sections 5-7 then deal with special cases, applications, and asymptotic questions that arise out of our central results, including a discussion on a theorem by Maniar and Nafiri [21] and how it follows as a simple corollary of our more general quantified asymptotics result.…”