Integral operators mapping into the space of bounded analytic functions

Abstract

Abstract. We address the problem of studying the boundedness, compactness and weak compactness of the integral operators Tg(f )(z) = z 0 f (ζ)g ′ (ζ) dζ acting from a Banach space X into H ∞ . We obtain a collection of general results which are appropriately applied and mixed with specific techniques in order to solve the posed questions to particular choices of X.

Unified Approach to Spectral Properties of Multipliers

“…ξ,k → 0, m ∈ Z ≥0 , uniformly on the sets A δ as k → ∞ is a consequence of the definition of g ξ,k . Moreover, the sequence (g ξ,k ) ∞ k=1 is a weak * null sequence by using Lemma 3.1 in [12]. Since (g ξ,k ) k ⊂ P(D) ⊂ B 0,α (D), we conclude that g ξ,k → 0 weakly when k → ∞.…”

Linear differential equations with slowly growing solutions

“…see [2,3,7,30,36]. Finally, we turn to consider coefficient conditions which place solutions of (1.2) in the Hardy spaces.…”

Intrinsic Operators from Holomorphic Function Spaces to Growth Spaces

“…The proof of the compactness can be found, for example, in [12,Lemma 3.7]. The proof of the weak compactness follows similarly, after using the Eberlein-Šmulian theorem (see [5]). Lemma 3.1.…”

“…For more details and more examples of this kind for the weighted composition operators or for the integral operators with X being either the space BM OA, V M OA, the Besov, or the Bloch spaces, see further [4] and [5].…”

“…To address further the compactness and weak compactness of intrinsic operators T : X → H v (B v ), we look at a special class of initial spaces. The extra conditions on the space X imposed in the next few theorems are part of the restrictions on the "admissible" domains considered in [4] and [5] for the weighted composition operators and the integral operators. We start first by considering a special kind of intrinsic weakly compact operators.…”