The Virozub-Matsaev condition and spectrum of definite type for self-adjoint operator functions
type: Item
instance of: scholarly article
title: The Virozub-Matsaev condition and spectrum of definite type for self-adjoint operator functions
zbMATH Open document ID: 1177.47019
author: Matthias Langer
author: A. Markus
author: Christiane Tretter
author: Heinz Langer
published in: Complex Analysis and Operator Theory
publication date: 2008-09-04T00:00:00Z
review text: Let \(T:{\mathcal D}\to {\mathcal B}({\mathcal H})\) be a self-adjoint operator function, that is, \(T(\lambda^*)=T(\lambda)^*,\;\lambda\in {\mathcal D}\), where \({\mathcal D}\) is some subset of the complex plane, symmetric with respect to \(\mathbb{R}\) and \({\mathcal B}({\mathcal H})\) is the space of bounded linear operators in a Hilbert space \({\mathcal H}\). For the continuously differentiable self-adjoint operator function \(T\), the Virozub-Matsaev condition (VM) is said to hold in a point \(\lambda_0\in {\mathcal D}\cap \mathbb{R}\) [on an interval \(\Delta\subset {\mathcal D}\cap \mathbb{R}\), resp.] if there exist positive constants \(\varepsilon\) and \(\delta\) such that \[ \|x\|=1,\;|(T(\lambda_0)x,x)|\leq\varepsilon\Longrightarrow (T'(\lambda_0)x,x)\geq \delta\tag{*} \] [if (*) holds for each \(\lambda_0\in\Delta\), resp.]. The condition (VM) is closely connected with the notion of a spectral point of positive type of the operator function \(T\) and to the existence and some properties of a local spectral function \(Q_t\) of \(T\) on an interval \(\Delta\) subject to \(\int_\Delta T(t)\,dQ_t=0\). Here, \(\lambda_0\in\sigma(T)\) is of positive type if (*) is in force with \(\|T(\lambda_0)x\|\) in place of \(|(T(\lambda_0)x,x)|\) therein. Following the notion and some results of the first and third authors and \textit{V.\,Matsaev} [J.~Funct.\ Anal.\ 235, No.\,1, 193--225 (2006; Zbl 1125.47010)], the authors of the present paper establish sufficient conditions for the condition (VM) to hold for twice continuously differentiable \(T\). Among others, the numerical range \(W(T)\) of \(T\) and its real subset \(W(T)\cap \mathbb{R}\) in conjunction with the so-called condition (vm), which is weaker and much simpler than (VM), play a key role. The following theorem is one of the main results. Let \(T\) be twice continuously differentiable on an open interval \(I=(\mu,\nu)\subset \mathbb{R}\) with respect to the operator norm, and let \(\omega(I)\in \bar{I}\…
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47B50
zbMATH DE Number: 5320870
zbMATH Keywords: selfadjoint operator function
zbMATH Keywords: numerical range
zbMATH Keywords: spectrum
MaRDI profile type: Publication
full work available at URL: s11785-007-0032-z
OpenAlex ID: W1972157696
DOI: 10.1007/S11785-007-0032-Z
description: scientific article; zbMATH DE number 5320870