Compressions of resolvents and maximal radius of regularity

Abstract

Suppose that λ − T \lambda - T is left invertible in L ( H ) L(H) for all λ ∈ Ω \lambda \in \Omega , where Ω \Omega is an open subset of the complex plane. Then an operator-valued function L ( λ ) L(\lambda ) is a left resolvent of T T in Ω \Omega if and only if T T has an extension T ~ \tilde {T} , the resolvent of which is a dilation of L ( λ ) L(\lambda ) of a particular form. Generalized resolvents exist on every open set U U , with U ¯ \overline {U} included in the regular domain of T T . This implies a formula for the maximal radius of regularity of T T in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by J. Zemánek is obtained.