Totally P-posinormal operators are subscalar

Let H be a complex infinite-dimensional separable Hilbert space. An operator T in L(H) is called totally P-posinormal (see [9]) iff there is a polynomial P with zero constant term such that ||P(T*z)h|| <= M(z) ||T(z)h|| for each h (element of) H, where Tz =T–zI and M(z) is bounded on the compacts of C. In this paper we prove that every totally P-posinormal operator is subscalar, i.e. it is the restriction of a generalized scalar operator to an invariant subspace. Further, a list of some important corollaries about Bishopʹs property (beta) and the existence of invariant subspaces is presented.