Conditions on the generator of a uniformly bounded \(C_0\)-semigroup
type: Item
instance of: scholarly article
title: Conditions on the generator of a uniformly bounded \(C_0\)-semigroup
zbMATH Open document ID: 0962.47019
DOI: 10.1007/BF02467113
publication date: 2001-06-18T00:00:00Z
review text: Suppose \(X\) is a complex Banach space with dual \(X^*\). Let \(A\) be a closed, densely defined operator on \(X\) with spectrum in the closed left half plane. The author proves that under the integral condition \[ \sup_{\delta> 0} \Biggl\{\delta \int^{\delta+ i\infty}_{\delta- i\infty}|\langle R^2(A, \lambda)x,y\rangle||d\lambda|\Biggr\}< \infty,\quad \forall x\in X,\;\forall y\in X^*, \] the operator \(A\) generates a uniformly bounded \(C_0\)-semigroup of operators. He also shows that this condition is not necessary in general Banach spaces, but for Hilbert spaces it is also necessary. The short and elegant proof is based on the Banach-Steinhaus theorem and the Miyadera-Feller-Phillips generation theorem. This result nicely complements the characterization of self-adjoint operators on a Hilbert space by \textit{J. A. van Casteren} [Pro. J. Math. 104, 241-255 (1983; Zbl 0501.47001)], see also Theorem 3.3 in the reviewer's paper, Collect. Math. 45, No. 3, 287-300 (1994; Zbl 0819.47018).
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47B40
zbMATH DE Number: 1452678
zbMATH Keywords: spectral operator
zbMATH Keywords: semigroup generating theorem
zbMATH Keywords: uniformly bounded \(C_0\)-semigroup of operators
zbMATH Keywords: Banach-Steinhaus theorem
zbMATH Keywords: Miyadera-Feller-Phillips generation theorem
author: A. M. Gomilko
reviewed by: Khristo N. Boyadzhiev
MaRDI profile type: Publication
cites work: A criterion for the similarity of a closed operator to a self-adjoint operator
cites work: Q5531981
cites work: Q3874458
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1452678