Power-regular Bishop operators and spectral decompositions

It is proved that a wide class of Bishop-type operators Tϕ,τ are power-regular operators in Lp(Ω,μ), 1⩽p<∞, computing the exact value of the local spectral radius at any function u∈Lp(Ω,μ). Moreover, it is shown that the local spectral radius at any u coincides with the spectral radius of Tϕ,τ as far as u is non-zero. As a consequence, it is proved that non-invertible Bishop-type operators are non-decomposable whenever log|ϕ|∈L1(Ω,μ) (in particular, not quasinilpotent); not enjoying even the weaker spectral decompositions \textit{Bishop property} (β) and \textit{property} (δ).