On a characterization of compactness and the Abel-Poisson summability of fourier coefficients in Banach spaces

Let $ T $ be the topological group of the unit circle with Euclidean topology, $ H $ be a complex Banach space, $ \alpha $ be a strongly continuous isometric linear representation of $ T $ in $ H $ , $ x\in H $ and $ \lbrace F_{k}^{\alpha}(x) \rbrace_{k \in\mathbb{Z}}$ be the family of Fourier coefficients of $ x $ with respect to $ \alpha $ . In this paper, an integral representation for Abel- Poisson mean operator of the family $ \lbrace F_{n}^{\alpha}(x)\rbrace $ of an $ x\in H $ is given, then by means of this representation it is proved that the family $ \lbrace F_{k}^{\alpha}(x) \rbrace_{k \in\mathbb{Z}} $ is Abel- Poisson summable to $x$ , and some tests for relatively compactness for a subset of $H$ are given in terms of $ \alpha $ .