k-Quasihyponormal operators are subscalar

In this paper we shall prove that if an operator T ∈ ℒ(⊕₁²H) is an operator matrix of the form T = (^{T1 T2} _{0 T3}) where T₁ is hyponormal and T₃^k = 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.