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 $L(\lambda) \coloneqq \lambda - A \pm B^* (C - \lambda )^{ - 1} B$ 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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