Integral operators mapping into the space of bounded analytic functions

TL;DR

This paper investigates the properties of integral operators mapping functions into the space of bounded analytic functions, focusing on boundedness, compactness, and weak compactness for various Banach spaces.

Contribution

It provides general criteria and techniques for analyzing the boundedness and compactness of integral operators into $H^$ for different Banach spaces $X$, advancing understanding in operator theory.

Findings

Derived conditions for boundedness of $T_g$

Established criteria for compactness and weak compactness

Applied results to specific Banach spaces

Abstract

We address the problem of studying the boundedness, compactness and weak compactness of the integral operators $T_g(f)(z)=\int_0^z f(\zeta)g'(\zeta)\,d\zeta$ acting from a Banach space $X$ into $H^\infty$. 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 $X$.