Semigroups of analytic Toeplitz operators on H2

For any function C analytic on the open unit disc, we define the Toeplitz operator T-C having domain D(T-C)={f is an element of H-2:Cf is in H-2} in the Hardy space H-2 by T-C:f-->Cf. Using a result of Suarez, we show that a closed operator R having domain D(R) dense in H-2 commutes with the standard unilateral shift S:f(z)-->zf(z) on D(R) if and only if there exits a function C in the Nevanlinna class N+ for which R=T-C on D(T-C) (see Corollary 3.3). Using this result, we show that a collection {Rt:t>0} of bounded linear operators commuting with S on H-2 is a C-0-semigroup if and only if there exists a function C analytic and having real part bounded above on the open unit disc for which R-t = T(e)tC for all t>0 (see Theorem 5.2).