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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 Tϕ,τ are power-regular operators in Lp(Ω,μ), 1, 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

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