Browder spectra of closed upper triangular operator matrices

Abstract

Let $ T_{B} = \left[\begin{array}{ll} A & B \\ 0 & D \end{array}\right] $ be an unbounded upper operator matrix with diagonal domain, acting in $ \mathcal H \oplus\mathcal K $, where $ \mathcal H $ and $ \mathcal K $ are Hilbert spaces. In this paper, some sufficient and necessary conditions are characterized under which $ T_{B} $ is Browder (resp., invertible) for some closable operator $ B $ with $ \mathcal D(B)\supset \mathcal D(D) $. Further, a sufficient and necessary condition is given under which the Browder spectrum (resp., spectrum) of $ T_{B} $ coincides with the union of the Browder spectrum (resp., spectrum) of its diagonal entries.

Keywords

• closed operator
• operator matrix
• browder spectrum
• spectrum