Scalar-type spectral operators and holomorphic semigroups
type: Item
instance of: scholarly article
title: Scalar-type spectral operators and holomorphic semigroups
zbMATH Open document ID: 0583.47040
DOI: 10.1007/BF02573198
author: R. deLaubenfels
published in: Semigroup Forum
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/134809
review text: We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold. (1) A generates a uniformly bounded holomorphic semigroup \(\{e^{- zA}\}_{Re(z)\geq 0}.\) (2) If \(F_ N(s)\equiv \int^{N}_{-N}\frac{\sin (sr)}{r}e^{irA}dr\), then \(\{\| F_ N\| \}^{\infty}_{N=1}\) is uniformly bounded on [0,\(\infty)\) and, for all x in X, the sequence \(\{F_ N(s)x\}^{\infty}_{N=1}\) converges pointwise on [0,\(\infty)\) to a vector-valued function of bounded variation. The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup \(\{e^{-zA}\}_{Re(z)\geq 0}\) generated by A, as follows. \[ \frac{1}{2}(E\{s\})x+(E[0,s))x=\lim_{N\to \infty}\int^{N}_{-N}\frac{\sin (sr)}{r}e^{irA}x\frac{dr}{\pi}, \] for any x in X.
Mathematics Subject Classification ID: 47B40
zbMATH DE Number: 3933847
zbMATH Keywords: scalar-type spectral operator
zbMATH Keywords: vector-valued function of bounded variation
zbMATH Keywords: projection-valued measure
MaRDI profile type: Publication
cites work: A characterization of scalar type operators on reflexive Banach spaces
cites work: Semi-groups of scalar type operators and a theorem of Stone
cites work: Q4164422
cites work: Q5654646
cites work: Characterization of unbounded spectral operators with spectrum in a half- line
cites work: Q5526485
cites work: Q4066072
cites work: Semigroups of Scalar Type Operators on Banach Spaces
description: scientific article; zbMATH DE number 3933847