The spectral mapping theorem for integrated semigroups
type: Item
instance of: scholarly article
title: The spectral mapping theorem for integrated semigroups
zbMATH Open document ID: 0781.47005
DOI: 10.1007/BF02573747
author: Yanyan Li
published in: Semigroup Forum
publication date: 1993-08-17T00:00:00Z
full work available at URL: https://eudml.org/doc/135260
review text: The article is devoted to one of the general theorems in the theory of strongly continuous semigroups -- the spectral mapping theorem. We denote as \(\sigma \bigl( T(t)\bigr)\) the spectrum of the semigroup \(\bigl( T(t) \bigr)_{t\geq 0}\), \(T:[0,\infty] \to L(E)\). The following main result for \(n\)-times integrated semigroups is represented in the article as Theorem 2. Let \(t\to S(t)\) \((t\geq 0)\) be an exponentially bounded \(n\)- times integrated semigroup. Then the spectral mapping theorem holds, i.e. \[ \sigma(S(t)) \cup\{0\}=\biggl\{ {1\over \lambda^ n}e^{\lambda t}- \sum^{n-1}_{k=0}{t^ k \over k! \lambda^{n-k}}| \lambda \in \sigma(A) \biggr\} \cup\{0\}. \]
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 270004
zbMATH Keywords: exponentially bounded \(n\)-times integrated semigroup
zbMATH Keywords: strongly continuous semigroups
zbMATH Keywords: spectral mapping theorem
reviewed by: Andreiy Kondrat'yev
MaRDI profile type: Publication
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cites work: Q3906660
cites work: Integrated semigroups and differential operators on \(L^ p\) spaces
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OpenAlex ID: W1977303792
description: scientific article