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Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization

Authors
  • Langer, Heinz1
  • Markus, Alexander2
  • Matsaev, Vladimir3
  • 1 Technische Universität Wien, Institut für Analysis und Scientific Computing, Wien, 1040, Austria , Wien (Austria)
  • 2 Ben-Gurion-University of the Negev, Department of Mathematics, Beer Sheva, 84105, Israel , Beer Sheva (Israel)
  • 3 School of Mathematical Sciences, Ramat Aviv, 69978, Israel , Ramat Aviv
Type
Published Article
Journal
Integral Equations and Operator Theory
Publisher
Birkhäuser-Verlag
Publication Date
Mar 25, 2009
Volume
63
Issue
4
Pages
533–545
Identifiers
DOI: 10.1007/s00020-009-1669-y
Source
Springer Nature
Keywords
License
Yellow

Abstract

In this note we continue the study of spectral properties of a self-adjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub–Matsaev condition on some interval Δ0 and is boundedly invertible in the endpoints of Δ0, then the ‘embedding’ of the original Hilbert space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document} into the Hilbert space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document}, where the linearization of A(z) acts, is in fact an isomorphism between a subspace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}(\Delta_{0})$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document}. As a consequence, properties of the local spectral function of A(z) on Δ0 and a so-called inner linearization of the operator function A(z) in the subspace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}(\Delta_{0})$$\end{document} are established.

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