A Note on Infinitesimal Generators of Semigroups on H2/φH2

Let φ be an inner function on the unit disc D. Let A(φ) be the commutant of the compression of the standard unilateral shift onto H2/φH2. Necessary and sufficient conditions for a complex number to be in the resolvent Set of the infinitesimal generator of a C0 semigroup of operators in A(φ) are given. In this case, the resolvents of the infinitesimal generator are obtained by solving a corona problem for unbounded functions whose real part is bounded above.