k-Quasihyponormal operators are subscalar

In this paper we shall prove that if an operator T is an element of L(+H-2(1)) is an operator matrix of the form T = (0T(1) T3T2) where T-1 is hyponormal and T-3(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 Prunarn (see [EP]). As a corollary we get that any k-quasihyponormal operators are subscalar.