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# On the Roughness of Quasinilpotency Property of One–parameter Semigroups

Published:2017-02-16
Printed: Jun 2017
• Ciprian Preda,
West University of Timişoara, Bd. V. Pârvan, No. 4, Timişoara 300223, Romania
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## Abstract

Let $\mathbf{S}:=\{S(t)\}_{t\geq0}$ be a C$_0$-semigroup of quasinilpotent operators (i.e. $\sigma(S(t))=\{0\}$ for each $t\gt 0$). In the dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical finite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a C$_0$-semigroup is preserved under the perturbations of its infinitesimal generator.
 Keywords: one-parameter semigroups, quasinilpotency, superstability, essential spectrum
 MSC Classifications: 34D05 - Asymptotic properties 34D10 - Perturbations 34E10 - Perturbations, asymptotics

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