Totally P-posinormal operators are subscalar

Let H be a complex infinite-dimensional separable Hilbert space. An operator T in L(R) is called totally P-posinormal (see [9]) iff there is a polynomial P with zero constant term such that parallel to(P) over bar (T*(z))hparallel to less than or equal to M(z)parallel toT(z)hparallel to for each h is an element of H, where T-z = 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 Bishop's property beta and the existence of invariant subspaces is presented.