On a Characterization of Compactness and the Abel-Poisson Summability of Fourier Coeffiients in Banach Spaces

Seda Öztürk

In this paper, for an isometric strongly continuous linear representation denoted by $\alpha$ of the topological group of the unit circle in complex Banach space, we study an integral representation for Abel-Poisson mean $A^\alpha_r(x)$ of the Fourier coefficients family of an element $x$, and it is proved that this family is Abel-Poisson summable to $x$. Finally, we give some tests which are related to characterizations of relatively compactness of a subset by means of Abel-Poisson operator $A^\alpha_r$ and $\alpha$.