Associated weights for spaces ofp-integrable entire functions

Abstract

In analogy to the notion of associated weights for weighted spaces of analytic functions with sup-norms, p-associated weights are introduced for spaces of entire p-integrable functions, 1 ≤ p < ∞. As an application, necessary conditions for the boundedness of composition operators acting between general Fock type spaces are proved. Notation and preliminariesConsider the space C N endowed with the complex inner product and the norm defined by:

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

“…In analogy to the notion of associated weights for weighted spaces of analytic functions with sup-norms, Mangino and the author introduced in [63] p-associated weights for spaces of entire p-integrable functions, 1 ≤ p < ∞. As an application, necessary conditions for the boundedness of composition operators acting between general Fock type spaces were proved.…”

Dynamics of Weighted Composition Operators on Spaces of Entire Functions of Exponential and Infraexponential Type

“…We are interested in the study of the dynamics of C w,ϕ when the multiplier is of the type w(z) = p N (z)e βz , p N a polynomial of degree N , and β ∈ C. The next result (see [10,Proposition 5] and [19,Proposition 3.1]) yields that for such multipliers, in order to have the continuity of C w,ϕ we must reduce to affine symbols, i.e. symbols of the form ϕ(z) = az +b for some a, b ∈ C. Proposition 3.…”

Dynamics of weighted composition operators on weighted Banach spaces of entire functions

“…< ∞, and so, w ≡ λ for some λ ∈ C. Then, if C w,ϕ = λC ϕ , the symbol must be affine by [2, Corollary 30] (see also [14,Proposition 3.1]). In particular:…”

“…In Section 3 we characterize the continuity and compactness of C w,ϕ on general weighted Banach spaces of entire functions, obtaining analogous results to those in [17] and [13] for the corresponding spaces on the disc and for unweighted composition operators, respectively. It is known (see [2,Corollary 30] and [14,Proposition 3.1]) that if the composition operator C ϕ on H vm (C) and on H 0 vm (C) is continuous, then the symbol ϕ must be affine, that is, ϕ(z) = az + b for some a, b ∈ C. In Proposition 6 we give a condition under which the continuity of the weighted composition operator also implies the affinity of the symbol. In Theorem 8 we characterize the continuity and compactness of C w,ϕ on H vm (C) and on H 0 vm (C) for affine symbols: In Section 4 we study the spectrum of the composition operator and the power boundedness, (uniform) mean ergodicity and hypercyclicity of C w,ϕ on the spaces H vm (C) and H 0 vm (C) when ϕ(z) = az + b.…”