A note about Volterra operators on weighted Banach spaces of entire functions

Abstract

We characterize boundedness, compactness and weak compactness of Volterra operators V g acting between different weighted Banach spaces H ∞ v (C) of entire functions with sup-norms in terms of the symbol g; thus we complement recent work by Bassallote, Contreras, Hernández-Mancera, Martín and Paul [3] for spaces of holomorphic functions on the disc and by Constantin and Peláez [16] for reflexive weighted Fock spaces.

The spectrum of Volterra-type integration operators on generalized Fock spaces

“…There is an extensive literature regarding the boundedness, compactness, and Schatten class membership of T g on various spaces of analytic functions (see [2,6] for Hardy spaces, [3,5,9,15,18,16] for weighted Bergman spaces, [12,13] for Dirichlet spaces, [10,11] for Fock spaces, [7,8] for growth spaces of entire functions, as well as the surveys [1,19] and references therein). One of the key tools in all these considerations is a Littlewood-Paleytype estimate for the target space of the operator, which, for a wide class of radial weights, can be obtained by standard techniques.…”

Boundedness of Volterra operators on spaces of entire functions

“…In this paper we would like to attack the boundedness of the Volterra operator V g (and certain modification of it) directly and not relying on the boundedness of the multiplication or differentiation operators independently. Note that the results in [12] do not apply to φ(r) = r α for 0 < α ≤ 2 and not cover different weights φ and ψ and the results in [9] cover different weights but only for p = q = ∞. We shall present here some necessary and sufficient conditions for the boundedness of V g from F φ p (C) into F ψ q (C) for different parameters 0 < p, q ≤ ∞ and different weights φ and ψ belonging to W, extending and providing some alternative proofs of some results in [9,11,12].…”

“…The study for F φ ∞ (C) = H ∞ v (C) was considered by Bonet and Taskinen [9] for certain classes of radial weights v. We refer also the interested reader to [8,11,13] for results concerning the spectra of the Volterra operator in this setting. In [9] certain class of weights J (see conditions appearing in [9, Proposition 3.2]) was introduced.…”

“…In [9] certain class of weights J (see conditions appearing in [9, Proposition 3.2]) was introduced. This class includes examples such as ψ(r) = βr α − γ log r − δ log(log(1 + r)), for some α, β > 0, γ, δ ∈ R, ψ(r) = (log(1 + r)) 1+ǫ − γ log r − δ log(log(1 + r)), for some ǫ > 0, γ, δ ∈ R or, more generally twice differentiable weights satisfying certain conditions (see [9,Thm 3.6,Thm 3.7]).…”

“…In [9] certain class of weights J (see conditions appearing in [9, Proposition 3.2]) was introduced. This class includes examples such as ψ(r) = βr α − γ log r − δ log(log(1 + r)), for some α, β > 0, γ, δ ∈ R, ψ(r) = (log(1 + r)) 1+ǫ − γ log r − δ log(log(1 + r)), for some ǫ > 0, γ, δ ∈ R or, more generally twice differentiable weights satisfying certain conditions (see [9,Thm 3.6,Thm 3.7]). For such a class, using the notationṽ(z) for the so-called associate weight of v (see [10]), they obtained (see [9,Theorem 3.4]) that for 0 = g ∈ H(C), v(z) = e −φ(|z|) and w(z) = e −ψ(|z|) with ψ ∈ J , the boundedness of…”