# Journal of Operator Theory

Volume 39, Issue 2, Spring 1998 pp. 339-359.

Spectral decomposition of some nonselfadjoint block operator matrices**Authors**: Heinz Langer (1), and Christiane Tretter (2)

**Author institution:**(1) Institut fur Analysis, Technische Mathematik und Versicherungsmathematik, Technische Universitaet Wien, Wiedner Hauptstr. 8--10, A--1040 Wien, Austria

(2) Naturwissenschaftliche Fakultat I -- Mathematik -- Universitaet Regensburg, Universitaetsstr. 31, D--93053 Regensburg, Germany

**Summary:**In this note we study spectral properties of a block operator matrix $\wt A$ (see (1.1) below), where $A$ and $-D$ are m-accretive, and $B, D$ are bounded operators. Under an additional assumption, the spectrum of $\wt A$ consists of one part in the extended right and one part in the left half plane, and the corresponding spectral subspaces allow representations by means of angular operators. If the part of the spectrum of $\wt A$ in the right half plane is discrete, a half range completeness statement follows. As an essential tool the quadratic numerical range of a block operator matrix is introduced.

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