Journal of Operator Theory

Volume 33, Issue 2, Spring 1995 pp. 259-277.

Spectral properties of a class of rational operator valued functions

Authors: Vadim M. Adamjan (1) and Heinz Langer (2)
Author institution:(1) Department of Theoretical Physics, Odessa University, Petra Velikogo 2, 701000 ODESSA, UKRAINE
(2) Institut für Analysis, Technische Mathematik und Versicherungsmathematik, Wiedner Hauptstrasse 8-10/114, A-1040 WIEN, AUSTRIA

Summary: We consider a selfadjoint operator function $L$ of the form $L(\lambda) := \lambda - A \pm B^* (C - \lambda )^{ - 1} B$ under the assumption that the spectrum of $L$ splits into two parts. In case of the sign + with the pencil $L$ there is associated a selfadjoint operator $\tilde A$ in some Hilbert space $\tilde{\mathcal H} \supset \mathcal H$, in case of the sign $-$ with $L$ there is associated a selfadjoint $\tilde B$ in a Kreĭn space $\tilde{\mathcal K} \supset \mathcal H$. Spectral properties of these associated operators are crucial for the study of the spectral properties of $L$. Sufficient conditions for the fact that the eigenvectors corresponding to certain parts of the spectrum of $L$ form a Riesz basis in $\mathcal H$ are given.

Keywords: Operator pencil, spectrum, eigenvector, Riesz basis.

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