Regular fractional weighted Wiener algebras and invariant subspaces

Abstract

Since the fifties, the interplay between spectral theory, harmonic analysis and a wide variety of techniques based on the functional calculus of operators, has provided useful criteria to find non-trivial closed invariant subspaces for operators acting on complex Banach spaces. In this article, some standard summability methods (mainly the Cesàro summation) are applied to generalize classical results due to Wermer [51] and Atzmon [8] regarding the existence of invariant subspaces under growth conditions on the resolvent of an operator. To do so, an extension of Beurling's regularity criterion [13] is proved for fractional weighted Wiener algebras $A_{\rho }^{\alpha }$ related with the Cesàro summation of order $\alpha \ge 0$. At the end of the article, other summability methods are considered for the purpose of finding new sufficient criteria which ensure the existence of invariant subspaces, resulting in several open questions on the regularity of fractional weighted Wiener algebras $A_{\rho }^{\mu }$ associated to matrix summation methods defined from non-vanishing complex sequences.