Generalized Volterra operators mapping between Banach spaces of analytic functions

Abstract

We characterize boundedness and compactness of the classical Volterra operator Tg : H ∞ vα → H ∞ induced by a univalent function g for standard weights vα with 0 ≤ α < 1, partly answering an open problem posed by A. Anderson, M. Jovovic and W. Smith. We also study boundedness, compactness and weak compactness of the generalized Volterra operator T ϕ g mapping between Banach spaces of analytic functions on the unit disc satisfying certain general conditions.2010 Mathematics Subject Classification. Primary 47B38, Secondary 46B50.

Volterra type operators on weighted Banach spaces of analytic functions

“…Here, C F is a constant depending only on function f . By the similar arguments from [9,20], we obtain the following two lemmas.…”

“…In this paper, using the ideas from [9,20], we give the sufficient and necessary conditions for the boundedness and compactness of T g :…”

Volterra Type Operators Between Bloch Type Spaces and Weighted Banach Spaces

“…Now we are to investigate the conditions for the boundedness of Volterra type operators T g and S g between Bloch type spaces B ∞ ν and B ∞ µ . Since the corresponding conditions for the boundedness and compactness of Volterra type operators T g between Bloch type spaces B ∞ ν and B ∞ µ had been studied in [12], thus, we concentrate on the companion operator S g . First, we prove the following lemma.…”

“…al. [12] studied the boundedness and compactness of T g between H ∞ να and H ∞ , where g is univalent and ν α = (1 − |z| 2 ) α with 0 ≤ α < 1.…”

“…In this paper, using the ideas from [12,26], we firstly give the sufficient and necessary conditions for the boundedness and compactness of T g : H ∞ ν → H ∞ µ when ν is analytic and log(g ′ ) ∈ B, and in the meantime, characterize the boundedness and compactness of S g : H ∞ ν → H ∞ µ when ν is analytic and log(g) ∈ B, where the weight µ is any weight and thus generalize their works. Then for some general weights, we characterize conditions for the boundedness and compactness of T g :…”