Vector-Valued Hausdorff Summability Methods and Ergodic Theorems

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Suppose $X$ and $Y$ are two general Banach spaces. Let $H = ({\Lambda _{n,k}})$ be a general ${\mathbf {B}}[X,Y]$-operator valued Hausdorff summability method: ${\Lambda _{n,k}} = (_k^n){\Delta ^{n - k}}{U_k}$ for $k \leq n$ and ${\Lambda _{n,k}} = {\theta _{X,Y}}$ for $k > n$, where $\{ {U_k}\} _{k = 0}^\infty$ is a sequence of operators in ${\mathbf {B}}[X,Y]$ and $\Delta$ denotes the backward difference (operator) and ${\theta _{X,Y}}(x) = {0_Y}$ (the zero element in $Y$) for all $x \in$. Then some necessary and sufficient conditions are given for the mean and uniform convergence of the averages \[ \sum \limits _{k = 0}^n {(_k^n){\Delta ^{n - k}}{U_k}({T^k}x)} \quad (x \in X,T \in {\mathbf {B}}[X]).\]