Dissipative Operators on Quotient Spaces of H2

Let φ be an inner function on the unit disc D. Necessary and sufficient conditions for an operator in the commutant A(φ) of the compression of the standard unilateral shift onto H2/φH2 to be dissipative are given. For a singular inner function φ, sufficient conditions for the Cayley transform of a constant multiple of a dissipative operator in A(φ) to be noncyclic and to belong to the class C0 of Sz.-Nagy and Foias are given.