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THE CESÀRO OPERATOR IN THE FRÉCHET SPACES ℓp+ AND Lp−

Part of: General theory of linear operators Topological linear spaces and related structures Special classes of linear operators

Published online by Cambridge University Press:  13 June 2016

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, e-mail: angela.albanese@unisalento.it
JOSÉ BONET
Affiliation:
Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, E-46071 València, Spain, e-mail: jbonet@mat.upv.es
WERNER J. RICKER
Affiliation:
Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, D-85072 Eichstätt, Germany e-mail: werner.ricker@ku.de
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Abstract

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The classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

MSC classification

Primary: 47A10: Spectrum, resolvent 47A16: Cyclic vectors, hypercyclic and chaotic operators 47A35: Ergodic theory
Secondary: 46A04: Locally convex Fréchet spaces and (DF)-spaces 47B34: Kernel operators 47B38: Operators on function spaces (general)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 273 - 287
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

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