Cesàro sums and algebra homomorphisms of bounded operators

Let X be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra l 1 (N 0 ) and fractional versions of Cesàro sums of a linear operator T ∈ B ( X ) is established. In particular, we show that every ( C , α )-bounded operator T induces an alge...

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Bibliographic Details
Published in:Israel journal of mathematics 2016-10, Vol.216 (1), p.471-505
Main Authors: Abadias, Luciano, Lizama, Carlos, Miana, Pedro J., Velasco, M. Pilar
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Banach spaces
Calculus
Convolution
Group Theory and Generalizations
Homomorphisms
Kernels
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operators (mathematics)
Sums
Theoretical
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Summary:Let X be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra l 1 (N 0 ) and fractional versions of Cesàro sums of a linear operator T ∈ B ( X ) is established. In particular, we show that every ( C , α )-bounded operator T induces an algebra homomorphism — and it is in fact characterized by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the ( C , α )-boundedness of the resolvent operator for temperated a-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-016-1417-3