Upper triangular matrix operators with diagonal (T_1,T_2), T_2 k-nilpotent

If (T=left(begin{array}{clcr}T_1&quad C 0&quad T_2end{array}right) in B(mathcal{X }_1oplus mathcal{X }_2)) is a Banach space upper triangular operator matrix with diagonal ((T_1, T_2)) such that (T_2) is (k)-nilpotent for some integer (kge 1), then (T) inherits a number of its spectral properties, such as SVEP, Bishop’s property ((beta )) and the equality of Browder and Weyl spectrum, from those of (T_1). This paper studies such operators. The conclusions are then applied to provide a general framework for results pertaining (for example) to Browder, Weyl type theorems and supercyclicity for classes of Hilbert space operators, such as (k)-quasi hyponormal, (k)-quasi isometric and (k)-quasi paranormal operators, defined by a positivity condition.