Existence of Non-subnormal Polynomially Hyponormal Operators

In 1950, P. R. Halmos, motivated in part by the successful development of the theory of normal operators, introduced the notions of subnormality and hyponormality for (bounded) Hilbert space operators. An operator T is subnormal if it is the restriction of a normal operator to an invariant subspace; T is hyponormal if T*T > TT*. It is a simple matrix calculation to verify that subnormality implies hyponormality, but the converse is false. One reason is that subnormality is invariant under polynomial calculus (indeed, analytic functional calculus), while hyponormality is not. If one then defines T to be polynomially hyponormal when p(T) is hyponormal for every polynomial p e C[z], the following question arises naturally.