The (Generalized) Weylness of Upper Triangular Operator Matrices

Abstract

Let ℋ and $${\cal K}$$ be complex infinite dimensional separable Hilbert spaces. We denote by $${M_C} = \left( {\begin{array}{*{20}{c}} A&C \\ 0&B \end{array}} \right)$$ a 2 × 2 upper triangular operator matrix acting on $${\cal H} \oplus {\cal K}$$ , where $$A \in {\cal B}\left({\cal H} \right),\,B \in {\cal B}\left({\cal K} \right)$$ and $$C \in {\cal B}\left({{\cal K},{\cal H}} \right)$$ . In this paper, we investigate the Weylness and generalized Weylness of MC for some (or every) $$C \in {\cal B}\left({{\cal K},{\cal H}} \right)$$ .