Power-regular Bishop operators and spectral decompositions

Abstract

It is proved that a wide class of Bishop-type operators $T_{\phi,\tau}$ are power-regular operators in $L^p(\Omega, \mu)$, $1 \leq p < \infty$, computing the exact value of the local spectral radius at any function $u \in L^p(\Omega, \mu)$. Moreover, it is shown that the local spectral radius at any $u$ coincides with the spectral radius of $T_{\phi,\tau}$ as far as u is non-zero. As a consequence, it is proved that non-invertible Bishop-type operators are non-decomposable whenever $\log|\phi| \in L^1(\Omega, \mu)$ (in particular, not quasinilpotent); not enjoying even the weaker spectral decompositions Bishop property $(\beta)$ and property $(\delta)$.