Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM)

H. Langer; A. Markus; V. Matsaev

doi:10.1007/978-3-0348-0221-5_20

Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM)

H. Langer, A. Markus, V. Matsaev

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

2 Scopus citations

Abstract

In this paper we continue the study of spectral properties of a selfadjoint analytic operator function A(z) under the Virozub-Matsaev condition. As in [6], [7], main tools are the linearization and the factorization of A(z). We use an abstract definition of a so-called Hilbert space linearization and show its uniqueness, and we prove a generalization of the well-known factorization theorem from [10]. The main results concern properties of the compression A_δ(z) of A(z) to its spectral subspace, called spectral compression of A(z). Close connections between the linearization, the inner linearization, and the local spectral function of A(z) and of its spectral compression A_δ(z) are established.

Original language	English
Title of host publication	A Panorama of Modern Operator Theory and Related Topics
Subtitle of host publication	The Israel Gohberg Memorial Volume
Editors	Harry Dym , Marinus A. Kaashoek, Peter Lancaster , Heinz Langer , Leonid Lerer
Publisher	Springer Basel
Pages	445-463
Number of pages	19
ISBN (Electronic)	9783034802215
ISBN (Print)	9783034802208, 9783034807890
DOIs	https://doi.org/10.1007/978-3-0348-0221-5_20
State	Published - 1 Jan 2012

Keywords

Factorization
Linearization
Self-adjoint analytic operator function
Spectral function
Spectrum of definite type

ASJC Scopus subject areas

General Mathematics

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10.1007/978-3-0348-0221-5_20

Cite this

Langer, H., Markus, A., & Matsaev, V. (2012). Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM). In H. Dym , M. A. Kaashoek, P. Lancaster , H. Langer , & L. Lerer (Eds.), A Panorama of Modern Operator Theory and Related Topics: The Israel Gohberg Memorial Volume (pp. 445-463). Springer Basel. https://doi.org/10.1007/978-3-0348-0221-5_20

Langer, H. ; Markus, A. ; Matsaev, V. / Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM). A Panorama of Modern Operator Theory and Related Topics: The Israel Gohberg Memorial Volume. editor / Harry Dym ; Marinus A. Kaashoek ; Peter Lancaster ; Heinz Langer ; Leonid Lerer . Springer Basel, 2012. pp. 445-463

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Langer, H, Markus, A & Matsaev, V 2012, Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM). in H Dym , MA Kaashoek, P Lancaster , H Langer & L Lerer (eds), A Panorama of Modern Operator Theory and Related Topics: The Israel Gohberg Memorial Volume. Springer Basel, pp. 445-463. https://doi.org/10.1007/978-3-0348-0221-5_20

Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM). / Langer, H.; Markus, A.; Matsaev, V.
A Panorama of Modern Operator Theory and Related Topics: The Israel Gohberg Memorial Volume. ed. / Harry Dym ; Marinus A. Kaashoek; Peter Lancaster ; Heinz Langer ; Leonid Lerer . Springer Basel, 2012. p. 445-463.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

TY - CHAP

T1 - Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM)

AU - Langer, H.

AU - Markus, A.

AU - Matsaev, V.

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N2 - In this paper we continue the study of spectral properties of a selfadjoint analytic operator function A(z) under the Virozub-Matsaev condition. As in [6], [7], main tools are the linearization and the factorization of A(z). We use an abstract definition of a so-called Hilbert space linearization and show its uniqueness, and we prove a generalization of the well-known factorization theorem from [10]. The main results concern properties of the compression Aδ(z) of A(z) to its spectral subspace, called spectral compression of A(z). Close connections between the linearization, the inner linearization, and the local spectral function of A(z) and of its spectral compression Aδ(z) are established.

AB - In this paper we continue the study of spectral properties of a selfadjoint analytic operator function A(z) under the Virozub-Matsaev condition. As in [6], [7], main tools are the linearization and the factorization of A(z). We use an abstract definition of a so-called Hilbert space linearization and show its uniqueness, and we prove a generalization of the well-known factorization theorem from [10]. The main results concern properties of the compression Aδ(z) of A(z) to its spectral subspace, called spectral compression of A(z). Close connections between the linearization, the inner linearization, and the local spectral function of A(z) and of its spectral compression Aδ(z) are established.

KW - Factorization

KW - Linearization

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KW - Spectral function

KW - Spectrum of definite type

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Langer H, Markus A, Matsaev V. Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM). In Dym H, Kaashoek MA, Lancaster P, Langer H, Lerer L, editors, A Panorama of Modern Operator Theory and Related Topics: The Israel Gohberg Memorial Volume. Springer Basel. 2012. p. 445-463 doi: 10.1007/978-3-0348-0221-5_20

Linearization, factorization, and the spectral compression of a self-adjoint analytic operator function under the condition (VM)

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Keywords

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