Weighted composition operators on the predual and dual Banach spaces of Beurling algebras on Z

Let vv be a weight sequence on ZZ and let ψ,φψ,φ be complex-valued functions on ZZ such that φ(Z)⊂Zφ(Z)⊂Z. In this paper we study the boundedness, compactness and weak compactness of weighted composition operators Cψ,φCψ,φ on predual Banach spaces c0(Z,1/v)c0(Z,1/v) and dual Banach spaces ℓ∞(Z,1/v)ℓ∞(Z,1/v) of Beurling algebras ℓ1(Z,v)ℓ1(Z,v).

On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

In this paper we provide the proof of a practical point-wise characterization of the set RP defined by the closure set of the real projections of the zeros of an exponential polynomial P(z) = Σn j=1 cjewjz with real frequencies wj linearly independent over the rationals. As a consequence, we give a complete description of the set RP and prove its invariance with respect to the moduli of the c′ js, which allows us to determine exactly the gaps of RP and the extremes of the critical interval of P(z) by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.

On Gruenhage spaces, separating σ-isolated families, and their relatives

We prove that, given a topological space X, the following conditions are equivalent. (α) X is a Gruenhage space. (β) X has a countable cover by sets of small local diameter (property SLD) by F∩G sets. (γ) X has a separating σ-isolated family M⊂F∩G. (δ) X has a one-to-one continuous map into a metric space which has a σ-isolated base of F∩G sets. Besides, we provide an example which shows Fragmentability ⇏ property SLD ⇏ the space to be Gruenhage.