Journal of Operator Theory

Volume 85, Issue 2, Spring 2021 pp. 403-416.

Power-regular Bishop operators and spectral decompositions

Authors: Eva A. Gallardo-Gutierrez (1), Miguel Monsalve-Lopez (2)
Author institution: (1) Departamento de Analisis Matematico y Matematica Aplicada, Facultad de Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias No 3, 28040 Madrid, Spain \textit{and} Instituto de Ciencias Matematicas ICMAT (CSIC-UAM-UC3M-UCM), Madrid, Spain
(2) Departamento de Analisis Matematico y Matematica Aplicada, Facultad de Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias No 3, 28040 Madrid, Spain and Inst. de Ciencias Matematicas ICMAT (CSIC-UAM-UC3M-UCM), Madrid, Spain

Summary: It is proved that a wide class of Bishop-type operators $T_{\phi,\tau}$ are power-regular operators in $L^p(\Omega,\mu)$, $1\leqslant p< \infty$, 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 \textit{Bishop property} ($\beta$) and \textit{property} ($\delta$).

DOI: http://dx.doi.org/10.7900/jot.2019sep21.2256
Keywords: power-regular operators, Bishop operators, decomposable operators

Contents Full-Text PDF