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SPECTRUM AND COMPACTNESS OF THE CESÀRO OPERATOR ON WEIGHTED $\ell _{p}$ SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices General theory of linear operators Special classes of linear operators

Published online by Cambridge University Press:  19 August 2015

ANGELA A. ALBANESE ,
JOSÉ BONET  and
WERNER J. RICKER
ANGELA A. ALBANESE*
Affiliation:
Dipartimento di Matematica e Fisica, ‘E. De Giorgi’, Università del Salento-C.P.193, I-73100 Lecce, Italy email [email protected]
JOSÉ BONET
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada, IUMPA, Universitat Politècnica de València, E-46071 Valencia, Spain email [email protected]
WERNER J. RICKER
Affiliation:
Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, D-85072 Eichstätt, Germany email [email protected]
*
✉[email protected]
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Abstract

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An investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator $\mathsf{C}$ when acting on the weighted Banach sequence spaces $\ell _{p}(w)$, $1<p<\infty$, for a positive decreasing weight $w$, thereby extending known results for $\mathsf{C}$ when acting on the classical spaces $\ell _{p}$. New features arise in the weighted setting (for example, existence of eigenvalues, compactness) which are not present in $\ell _{p}$.

Keywords

Cesàro operatorweighted ℓp spacespectrumcompact operator

MSC classification

Primary: 47A10: Spectrum, resolvent 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Secondary: 46B45: Banach sequence spaces 47A16: Cyclic vectors, hypercyclic and chaotic operators 47A30: Norms (inequalities, more than one norm, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 99 , Issue 3 , December 2015 , pp. 287 - 314
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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