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 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
Mathematics Subject Classification (2000). 47A56, 47A68, 47A10.
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References
B. Čurgus, A. Dijksma, H. Langer, H.S.V. de Snoo: Characteristic functions of unitary colligations and of bounded operators in Krein spaces. Operator Theory: Adv. Appl. 41 (1989), 125–152.
I. Gohberg, M.A. Kaashoek, D.C. Lay: Equivalence, linearization and decomposition of holomorphic operator functions. J. Funct. Anal. 28 (1978), 102–144.
M.A. Kaashoek, C.V.M. van der Mee, L. Rodman: Analytic operator functions with compact spectrum. I. Spectral nodes, linearization and equivalence. Integral Equations Operator Theory 4 (1981), 504–547.
H. Langer, A. Markus, V. Matsaev: Locally definite operators in indefinite inner product spaces. Math. Ann. 308 (1997), 405–424.
H. Langer, A. Markus, V. Matsaev: Linearization and compact perturbation of selfadjoint analytic operator functions.Operator Theory:Adv.Appl. 118 (2000), 255–285.
H. Langer, A. Markus, V. Matsaev: Self-adjoint analytic operator functions and their local spectral function. J. Funct. Anal. 235 (2006), 193–225.
H. Langer, A. Markus, V. Matsaev: Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization. Integral Equations Operator Theory 63 (2009), 533–545.
A.S. Markus: Introduction to the Spectral Theory of Polynomial Operator Pencils. AMS Translations of Mathematical Monographs, vol. 71, 1988.
A. Markus, V. Matsaev: On the basis property for a certain part of the eigenvectors and associated vectors of a self-adjoint operator pencil. Math. USSR Sbornik 61 (1988), 289–307.
A.I. Virozub, V.I. Matsaev: The spectral properties of a certain class of selfadjoint operator-valued functions. Funct. Anal. Appl. 8 (1974), 1–9.
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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: Dym, H., Kaashoek, M., Lancaster, P., Langer, H., Lerer, L. (eds) A Panorama of Modern Operator Theory and Related Topics. Operator Theory: Advances and Applications(), vol 218. Springer, Basel. https://doi.org/10.1007/978-3-0348-0221-5_20
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DOI: https://doi.org/10.1007/978-3-0348-0221-5_20
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