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

Convex-Cyclic Weighted Composition Operators and Their Adjoints

“…The study of the weighted composition operator W (u,ψ) : f → u • f (ψ) with symbol ψ and multiplier u acting on various spaces of holomorphic functions traces back to works related to isometries on the Hardy spaces [11,14] and commutants of Toeplitz operators [8,9]. Since then, the operator has attracted much research interest and there exists now rich body of literatures dealing with many of its properties in various settings; see for example [2,7,16,18] and the references therein.…”

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

“…The first space is formed by the entire functions which are of exponential type α for some α > 0, endowed with its natural locally convex topology which makes it an (LB)-space, and the second one by all the entire functions which are of exponential type for each α > 0, endowed with its natural locally convex topology which makes it a Fréchet space. Here we continue the research of the first author in [10], where the dynamics of the operator is studied on weighted Banach spaces of entire functions H v α , H 0 v α , defined by weights of exponential type v α (z) = e −αz , α > 0, z ∈ C. We refer to the next section for the precise notation and definitions.…”

“…3 we characterize the continuity of the operator when the symbol ϕ is an affine function, that is, when ϕ(z) = az + b, a, b ∈ C, and we show it is never compact. In the setting of the Banach spaces H v α and H 0 v α , the operator C w,ϕ is not continuous if |a| > 1, or if |a| = 1 and the multiplier w is not constant [10,Theorem 8]. On the spaces Exp and Exp 0 , for every a ∈ C we obtain continuity for multipliers of the form w(z) = p(z)e βz , β ∈ C, in the case of Exp and w(z) = p(z) in the case of Exp 0 , p being a polynomial.…”

“…4. Since the pioneer work of Bonet and Domański [18], the study of ergodic properties of composition operators and weighted composition operators in Banach and Fréchet spaces of analytic functions has become a very active area of research in mathematical analysis, see [7,8,10,12,13,23,28]. For natural weighted composition operators we completely characterize when C w,ϕ is power bounded and (uniformly) mean ergodic on Exp and on Exp 0 .…”

“…is power bounded and mean ergodic for each α > α 0 (α > 0) except in the case of Theorem ME-Exp (ii), where if β = 0, there is no α > 0 such that C w,ϕ (H 0 v α ) ⊆ H 0 v α , as w is not constant (see [10,Theorem 8]). We point out that the theorems in Sect.…”