Multipliers on Weighted Sequence Spaces

The space $\ell^{p,\alpha}$ of complex sequences $\mathbf{a} = (a_{0},a_{1},a_{2},\ldots)$ for which \[ \|\mathbf{a}\|_{p,\alpha} = \left( \sum_{k=0}^{\infty} |a_{k}|^{p} (k+1)^{\alpha} \right)^{1/p} \lt \infty \] is studied. Each such sequence can be identified with the analytic function with power series \[ f(z) = \sum_{k=0}^{\infty} a_{k} z^{k}. \] In this setting, the point evaluation and the difference quotient mappings are shown to be bounded; the cases are identified in which $\ell^{p,\alpha}$ is boundedly contained in $\ell^{r,\beta}$. Conditions on the parameters are derived for the analytic functions of $\ell^{p,\alpha}$ to have radial limits almost everywhere on the boundary, and for $\ell^{p,\alpha}$ to be an algebra. Smoothness properties of the boundary function are investigated. Basic properties of multipliers on $\ell^{p,\alpha}$ are established, and conditions on the multiplier norm and coefficient growth are derived. Multipliers having a certain extremal property are described. A discrete version of the Schur Test is obtained, and used to produce a family of examples of multipliers.