Compressions of resolvents and maximal radius of regularity

Abstract:

Suppose that $\lambda - T$ is left invertible in $L(H)$ for all $\lambda \in \Omega$, 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 $\tilde {T}$, the resolvent of which is a dilation of $L(\lambda )$ of a particular form. Generalized resolvents exist on every open set $U$, with $\overline {U}$ included in the regular domain of $T$. This implies a formula for the maximal radius of regularity of $T$ in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by J. Zemánek is obtained.