k-Quasihyponormal operators are subscalar

Abstract

In this paper we shall prove that if an operator T ∈ ℒ(⊕12H) is an operator matrix of the form T = (T1 T20 T3) where T1 is hyponormal and T3k = 0, then T is subscalar of order 2(k + 1). Hence non-trivial invariant subspaces are known to exist if the spectrum of T has interior in the plane as a result of a theorem of Eschmeier and Prunaru (see [EP]). As a corollary we get that any k-quasihyponormal operators are subscalar.