The Brauer–Ostrowski Theorem for Matrices of Operators

Abstract

The classical Brauer-Ostrowski Theorem gives a localization of the spectrum of a matrix by a union of Cassini ovals. In this paper we prove a corresponding result for operator matrices. (2000). Primary 47A10; Secondary 47A05. Mathematics Subject Classification

A new Gershgorin-type result for the localisation of the spectrum of matrices

“…Cassini ovals for operator matrices of bounded linear operators are studied in [, § 5]. It is easy to check that the proof in [, Thm. 5.1] remains valid if the diagonal entries are merely closed.…”

“…However, it is not clear whether it can be adapted to the case of unbounded off-diagonal entries. Furthermore, Cassini-type inclusions have been proved in [HS07] merely for the approximate point spectrum of such operator matrices: We are going to sharpen said spectral localisation as a consequence of the results in the previous section. Definition 4.7.…”

“…It, too, can be partially extended to general operator matrices of bounded linear operators. This has been done in [, § 5]. So far, Cassini‐type inclusions have been proved merely for the approximate point spectrum of such operator matrices.…”