Compressions of resolvents and maximal radius of regularity

Suppose that lambda - T is left invertible in L(H) for all lambda is an ele ment of Omega, where Omega is an open subset of the complex plane. Then an operator-valued function L(lambda) is a left resolvent of T in Omega if and only if T has an extension (T) over tilde, the resolvent of which is a dil ation of L(lambda) of a particular form. Generalized resolvents exist on ev ery open set U, with (U) over bar included in the regular domain of T. This implies a formula for the maximal radius of regularity of T in terms of th e spectral radius of its generalized inverses. A solution to an open proble m raised by J. Zemanek is obtained.