Posinormal composition operators on H2

Abstract

A bounded linear operator A on a Hilbert space is posinormal if there exists a positive operator P such that AA⁎=A⁎PA. Posinormality of A is equivalent to the inclusion of the range of A in the range of its adjoint A⁎. Every hyponormal operator is posinormal, as is every invertible operator. We characterize both the posinormal and coposinormal composition operators Cφ on the Hardy space H2 of the open unit disk D when φ is a linear-fractional selfmap of D. Our work reveals that there are composition operators that are both posinormal and coposinormal yet have powers that fail to be posinormal.