Journal of Operator Theory

Volume 39, Issue 1, Winter 1998 pp. 3-41.

On the discrete spectrum of some selfadjoint operator matrices

Authors: V. Adamyan (1), R. Mennicken (2), and J. Saurer (3)
Author institution: (1) Odessa University, Department of Theoretical Physics, Dvorjanskaja 2, Odessa 270100, Ukraine
(2) University of Regensburg, Department of Mathematics, D-93040 Regensburg, Germany
(3) University of Regensburg, Department of Mathematics, D-93040 Regensburg, Germany

Summary: This paper is devoted to the study of the discrete spectrum of selfadjoint operators, which are generated by symmetric operator matrices of the form $$\L_0 = \pmatrix{A & B \cr B^\ast & C}$$ in the product Hilbert space $\H_1\times\H_2$, where the entries $A$, $B$ and $C$ are not necessarily bounded operators in the Hilbert spaces $\H_1$, $\H_2$ or between them, respectively. Under some assumptions all selfadjoint extensions of $\L_0$ in $\H_1\times \H_2$ are described and the extension $\L$ defined by the given selfadjoint operator $C$ is singled out. General statements on the discrete spectrum of $\L$ and its accumulation points are proved. Special attention is paid to the case that C is bounded.

Keywords: Selfadjoint operator matrices, discrete spectrum, accumulation points, matrix differential operators

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