Ces\'aro sums and algebra homomorphisms of bounded operators

Abstract

Let $X$ be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra $\ell^{1}(\mathbb{N}_0)$ and the algebraic structure of Ces\`{a}ro sums of a linear operator $T\in \mathcal{B}(X)$ is established. In particular, we show that every $(C, \alpha)$-bounded operator $T$ induces - and 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 $(C,\alpha)$ boundedness of the resolvent operator for temperated $\alpha$-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.