k-Quasihyponormal operators are subscalar

Abstract

In this paper we shall prove that if an operatorT∈L(⊕ ²₁ H) is an operator matrix of the form

$$T = \left( {\begin{array}{*{20}c} {T_1 } & {T_2 } \\ 0 & {T_3 } \\ \end{array} } \right)$$

whereT ₁ is hyponormal andT ^k₃ =0, thenT is subscalar of order 2(k+1). Hence non-trivial invariant subspaces are known to exist if the spectrum ofT has interior in the plane as a result of a theorem of Eschmeier and Prunaru (see [EP]). As a corollary we get that anyk-quasihyponormal operators are subscalar.