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

摘要

If T, is a Banach space upper triangular operator matrix with diagonal (T_1, T_2) such that T_2 is k-nilpotent for some integer k ≥ 1, then T inherits a number of its spectral properties, such as SVEP, Bishop's property (β) 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.