Local semigroups of isometries in \(\Pi_ \kappa\)-spaces and related continuation problems for \(\kappa\)-indefinite Toeplitz kernels
type: Item
instance of: scholarly article
title: Local semigroups of isometries in \(\Pi_ \kappa\)-spaces and related continuation problems for \(\kappa\)-indefinite Toeplitz kernels
zbMATH Open document ID: 0778.47024
DOI: 10.1007/BF01195778
author: R. Bruzual
author: Stefania A. M. Marcantognini
published in: Integral Equations and Operator Theory
publication date: 1993-01-16T00:00:00Z
review text: Let \(I=\mathbb{R}\) or \(I=[-a,a]\) with \(0<a<\infty\). An Hermitian kernel \(F(x,y)\) on \(I\) is said to be \(\kappa\)-indefinite with \(\kappa<\infty\) if for any \(n\) and any points \(x_ 1,\dots,x_ n\) in \(I\) the \(n\times n\) matrix \([F(x_ i,x_ j)]\) has at most \(\kappa\) negative eigenvalues, and exactly \(\kappa\) for some choice. \(F\) is called a Toeplitz kernel if \(F(x,y)=f(x-y)\) for some continuous function \(f\) on the closure of \(I-I\), while it is called a generalized Toeplitz kernel (G.T.K.) if there are four continuous functions \(f_{\alpha\beta}\) (\(\alpha,\beta=1,2\)) on the closure of \(I_ \alpha-I_ \beta\) such that \(F(x,y)=f_{\alpha\beta}(x-y)\) for \(x\in I_ \alpha\), \(y\in I_ \beta\), where \(I_ 1=I\cap[0,+\infty)\) and \(I_ 2=I\cap(- \infty,0]\). \textit{M. G. Krein} and \textit{H. Langer} [J. Operator Theory 13, 299-417 (1985; Zbl 0608.47038)] proved that every \(\kappa\)-indefinite Toeplitz kernel on \(I=[-a,a]\) can be extended to a \(\kappa\)-indefinite Toeplitz kernel on \(\mathbb{R}\). In the present paper the authors generalize the Krein-Langer theorem to the case of \(\kappa\)-indefinite G.T.K. Considering a local semigroup of isometries on a Pontryagin space with generator \(F\) they reduce the problem to the assertion that every strongly continuous local semigroup of isometries can be extended to a unitary group on larger Pontryagin space with same index.
reviewed by: T. Ando
Mathematics Subject Classification ID: 47B50
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 42A82
zbMATH DE Number: 92245
zbMATH Keywords: indefinite kernel
zbMATH Keywords: Hermitian kernel
zbMATH Keywords: Toeplitz kernel
zbMATH Keywords: Krein-Langer theorem
zbMATH Keywords: strongly continuous local semigroup of isometries
zbMATH Keywords: Pontryagin space with same index
MaRDI profile type: Publication
cites work: Q4774569
cites work: Local semigroups of contractions and some applications to Fourier representation theorems
cites work: Q3894416
cites work: Über indexerhaltende Erweiterungen eines hermiteschen Operators im Pontrjaginraum
cites work: Q4743338
cites work: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume \(\Pi_k\). zusammenhängen. II. Verallgemeinerte Resolventen, u-Resolventen und ganze Operatoren
cites work: Defect subspaces and generalized resolvents of an Hermitian operator in the space \(\Pi_\kappa\)
cites work: Q5533814
cites work: Q3961739
cites work: Q3478990
cites work: Q5600360
full work available at URL: https://doi.org/10.1007/bf01195778
OpenAlex ID: W2047697178
Recommended article: On the extension problem for continuous positive definite generalized Toeplitz kernels definite on a finite interval
Recommended article: The Kreĭn-Langer problem for Hilbert space operator valued functions on the band
Recommended article: Q3472696
Recommended article: Q3469715
Recommended article: On the Relation Between Positive Definite Functions and Generalized Toeplitz Kernels
Recommended article: On extensions of indefinite functions defined on a rectangle
Recommended article: Q2915382
Recommended article: Q2943681
Recommended article: Q5448145
Recommended article: Q5185494
description: scientific article; zbMATH DE number 92245