Spectral inclusion for unbounded block operator matrices
type: Item
instance of: scholarly article
title: Spectral inclusion for unbounded block operator matrices
zbMATH Open document ID: 1179.47005
author: Christiane Tretter
published in: Journal of Functional Analysis
publication date: 2009-06-17T00:00:00Z
review text: The quadratic numerical \(W^2(\mathcal A)\) range of the block operator matrix \[ \mathcal A = \begin{pmatrix} A & B \cr C & D\cr \end{pmatrix}, \qquad \mathcal D(\mathcal A) = \mathcal D(A)\cap\mathcal D(C) \oplus \mathcal D(B)\cap\mathcal D(D) \subseteq \mathcal H_1\oplus \mathcal H_2, \] is the set of all eigenvalues of all \(2\times 2\) matrices \[ \mathcal A_{f,g} = \begin{pmatrix} (Af,f) & (Bg,f) \cr (Cf,g) & (Dg,g)\cr \end{pmatrix} \in M_2(\mathbb C), \] where \(f\in \mathcal D(A)\cap\mathcal D(C)\), \(g\in \mathcal D(B)\cap\mathcal D(D)\) with \(\|f\| = \|g\| = 1\) and \(\mathcal H_1,\mathcal H_2\) are Hilbert spaces. It is assumed that \(A, B, C\) and \(D\) are closed and densely defined linear operators. In contrast to the numerical range \(W(\mathcal A)\), the quadratic numerical range is not necessarily convex. In the main part of the article, the author proves the spectral inclusions \[ \sigma_p(\mathcal A) \subseteq W^2(\mathcal A) \subseteq W(\mathcal A), \quad \sigma_{app}(\mathcal A) \subseteq \overline{ W^2(\mathcal A) } \subseteq \overline{ W(\mathcal A)} \] if \(\mathcal A\) is diagonally dominant or off-diagonally dominant of order \(0\). Conditions that guarantee that \(\mathcal A\) is diagonally dominant or off-diagonally dominant are given in the first part of the paper. The paper concludes with an application to off-diagonally dominant selfadjoint block operator matrices with separated diagonal elements (a typical example is the Dirac operator in \(\mathbb R^3\)) and block operator matrices where \(A\) and \(D\) are selfadjoint and \(C=-B^\ast\) or \(C=B^\ast\) (operators of the latter type appear as models for a two channel Hamiltonian). The paper generalizes results of \textit{H.\,Langer, A.\,Markus, V.\,Matsaev} and \textit{C.\,Tretter} [Linear Algebra Appl.\ 330, No.\,1--3, 89--112 (2001; Zbl 0998.15035)] and \textit{H.\,Langer} and \textit{C.\,Tretter} [J.~Oper.\ Theory 39, No.\,2, 339--359 (1998; Zbl 0996.47006)].
reviewed by: Monika Winklmeier
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A25
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 5565776
zbMATH Keywords: block numerical range
zbMATH Keywords: block operator matrices
zbMATH Keywords: spectral inclusion
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.12.024
OpenAlex ID: W1967281861
cites work: Q4865552
cites work: Bounds on variation of spectral subspaces under \(J\)-self-adjoint perturbations
cites work: The rotation of eigenvectors by a perturbation
cites work: The rotation of eigenvectors by a perturbation. II
cites work: The Rotation of Eigenvectors by a Perturbation. III
cites work: Q3764747
cites work: Eigenvalue estimates in the semi–classical limit for Pauli and Dirac operators with a magnetic field
cites work: Perturbation theory for linear operators.
cites work: On the existence of solutions to the operator Riccati equation and the \(\tan\Theta\) theorem
cites work: Perturbation of spectra and spectral subspaces
cites work: A new concept for block operator matrices: The quadratic numerical range
cites work: Q2752825
cites work: Q4529287
cites work: Some sharp norm estimates in the subspace perturbation problem
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q4331959
cites work: Q3873024
DOI: 10.1016/J.JFA.2008.12.024
description: scientific article; zbMATH DE number 5565776