On Weighted Sequence Spaces

The space ℓp,α of complex sequences a = (a0,a1,a2, . . .) for which ∞ ∥a∥p,α = ( Σ|ak|p(k+1)α)1/p < ∞ k=0 is studied. Each such sequence can be identified with the analytic function with power series ∞ f (z) = ∑ akzk. k=0 In this setting, the point evaluation and the difference quotient mappings are shown to be bounded; the cases are identified in which ℓp,α is boundedly contained in ℓr,β . Conditions on the parameters are derived for the analytic functions of ℓp,α to have radial limits almost everywhere on the boundary of the domain, and for ℓp,α to be an algebra. Smoothness properties of the boundary function are investigated. Basic properties of multipliers on ℓp,α 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.