Multiplicative operators on analytic function spaces

TL;DR

This paper investigates the conditions under which operators on various analytic function spaces are multiplicative or almost multiplicative, correcting previous results and providing new characterizations, especially for composition operators.

Contribution

It corrects and extends Schwartz's 1969 result on multiplicative operators on Hardy spaces and characterizes multiplicative composition operators on several classical spaces.

Findings

Operators are almost multiplicative iff they are composition operators on certain spaces.

Schwartz's original proof has a gap for $H^\infty$, which is addressed.

Complete characterization of multiplicative composition operators with respect to the Duhamel product.

Abstract

H. J. Schwartz proved in his thesis (1969) that a nonzero bounded operator on Hardy spaces $(H^p, 1\leq p\leq\infty)$ is almost multiplicative if and only if it is a composition operator. But, his proof has a gap. In this article, we show that his result is not correct for $H^\infty$ and we fill the gap for $H^p, 1\leq p<\infty.$ Further, we prove that on several classical spaces such as the Bloch space, the little Bloch space, Besov spaces $B_p$ for $p>1$, and weighted Bergman spaces an operator is almost multiplicative if and only if it is a composition operator. Finally, we give a complete characterization of those composition operators that are multiplicative with respect to the Duhamel product of analytic functions.