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

Part of: Special classes of linear operators Normed linear spaces and Banach spaces; Banach lattices General theory 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 angela.albanese@unisalento.it
JOSÉ BONET
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada, IUMPA, Universitat Politècnica de València, E-46071 Valencia, Spain email jbonet@mat.upv.es
WERNER J. RICKER
Affiliation:
Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, D-85072 Eichstätt, Germany email werner.ricker@ku-eichstaett.de
*
✉angela.albanese@unisalento.it
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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. 

References

Akhmedov, A. M. and Başar, F., ‘On the fine spectrum of the Cesàro operator in c 0’, Math. J. Ibaraki Univ. 36 (2004), 25–32.CrossRefGoogle Scholar
Akhmedov, A. M. and Başar, F., ‘The fine spectrum of the Cesàro operator C 1 over the sequence space bv p , (1 ≤ p < ∞)’, Math. J. Okayama Univ. 50 (2008), 135–147.Google Scholar
Albanese, A. A., Bonet, J. and Ricker, W. J., ‘Convergence of arithmetic means of operators in Fréchet spaces’, J. Math. Anal. Appl. 401 (2013), 160–173.CrossRefGoogle Scholar
Ansari, S. I. and Bourdon, P. S., ‘Some properties of cyclic operators’, Acta Sci. Math. (Szeged) 63 (1997), 195–207.Google Scholar
Bennett, G., ‘Some elementary inequalities’, Q. J. Math. 38 (1987), 401–425.CrossRefGoogle Scholar
Brown, A., Halmos, P. R. and Shields, A. L., ‘Cesàro operators’, Acta Sci. Math. (Szeged) 26 (1965), 125–137.Google Scholar
Curbera, G. P. and Ricker, W. J., ‘Spectrum of the Cesàro operator in ℓ p’, Arch. Math. 100 (2013), 267–271.CrossRefGoogle Scholar
Curbera, G. P. and Ricker, W. J., ‘Solid extensions of the Cesàro operator on ℓ p and c 0’, Integral Equations Operator Theory 80 (2014), 61–77.CrossRefGoogle Scholar
Curbera, G. P. and Ricker, W. J., ‘The Cesàro operator and unconditional Taylor series in Hardy spaces’, Integral Equations Operator Theory, to appear, doi:10.1007/s00020-015-2230-9.Google Scholar
Dunford, N. and Schwartz, J. T., Linear Operators I: General Theory, 2nd printing (Wiley Interscience Publications, New York, 1964).Google Scholar
Jameson, G. J. O., ‘Norms and lower bounds of operators on the Lorentz sequence space d (w, 1)’, Illinois J. Math. 43 (1999), 79–99.CrossRefGoogle Scholar
Jameson, G. J. O. and Lashkaripour, R., ‘Norms of certain operators on weighted ℓ p spaces and Lorentz sequences spaces’, J. Inequal. Pure Appl. Math. 3(Article 6) (2002), 17 pp.Google Scholar
Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities (Cambridge University Press, Cambridge, 1964).Google Scholar
Leibowitz, G., ‘Spectra of discrete Cesàro operators’, Tamkang J. Math. 3 (1972), 123–132.Google Scholar
Megginson, R. E., An Introduction to Banach Space Theory (Springer, New York, 1998).CrossRefGoogle Scholar
Mureşan, M., A Concrete Approach to Classical Analysis (Springer, Berlin, 2008).Google Scholar
Okutoyi, J. I., ‘On the spectrum of C 1 as an operator on bv 0’, J. Aust. Math. Soc. 48 (1990), 79–86.CrossRefGoogle Scholar
Reade, J. B., ‘On the spectrum of the Cesàro operator’, Bull. Lond. Math. Soc. 17 (1985), 263–267.CrossRefGoogle Scholar
Rhoades, B. E. and Yildirim, M., ‘Spectra for factorable matrices on ℓ p’, Integral Equations Operator Theory 55 (2006), 111–126.CrossRefGoogle Scholar