Compressions of Resolvents and Maximal Radius of Regularity

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 $\bar{U}$ 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. Zem\'anek is obtained.