Differentiation and integration operators on weighted Banach spaces of holomorphic functions

Abstract

We obtain a new natural description of the class of radial weights for which some previous results of A. Harutyunyan and W. Lusky concerning the boundedness of differentiation and integration operators on corresponding spaces are valid. To do this, we develop an elementary approach which is essentially different from the previous one and allows us to establish several new results and new characterizations of some popular classes of radial weights.

Integration and Differentiation in Hardy Growth Spaces

“…Next, for 0 < p < ∞, we introduce a notion of p-associated weight w p . In fact, using p-associated weights, we study the differentiation operator D defined on Hol(D) and show that several results from [3] are extendable to H p w (D) with 1 ≤ p < ∞ and partially with 0 < p < ∞. Also, we obtain related results for the integration operator J.…”

“…The bounded and compact operators J : H ∞ w (C) → H ∞ u (C) are studied in [2,3] under specific restrictions on w or for w = u. We show that the problems in question have quite elementary solutions if w is assumed to be equivalent to a power series 2010 Mathematics Subject Classification.…”

“…General conditions related to the boundedness of the differentiation operator D : H p w (D) → H p u (D) are presented in Section 4. Finally, as in [3,9], we consider in Section 5 the following specific situations: w(r) = (1 − r)u(r) for 0 ≤ r < 1 and w(r) = u(r) for 0 ≤ r < +∞.…”

Dynamics of the Volterra-Type Integral and Differentiation Operators on Generalized Fock Spaces

“…The differentiation operator D has been studied on Banach spaces of analytic functions by several authors. Harutyunyan and Lusky [17] identified conditions under which the operator becomes bounded when acting between weighted spaces of holomorphic functions endowed with the supremum norm; see also [1]. Bonet [8] studied various dynamical properties of the operator on these weighted spaces, and the study was continued jointly with Beltrán, Bonilla and Fernández in [6,10].…”

“…For holomorphic functions f and g, the differentiation operator Df = f ′ and the Volterra-type integral operator V g f (z) = ∫ z 0 g ′ (w)f (w)dw are classical objects in operator theory, function spaces and differential equations. Many of their basic properties including boundedness, compactness and spectra have been extensively studied when acting on several function spaces over various domains; see for example [1,5,6,8,10,11,13,20,21,22,23] and the references therein. Understanding the dynamical structures of these operators is another important and basic problem in operator theory.…”