Multiplicative operators on analytic function spaces

TL;DR

The paper revisits the classical question of when almost multiplicative operators on analytic function spaces are composition operators. It provides corrected proofs showing that for $H^p$ with $1\le p<\infty$ (and many classical spaces such as $B_p$, $\mathcal{B}$, $\mathcal{B}_0$, and $A^p_\alpha$) every nonzero bounded almost multiplicative operator is a composition operator, while Schwartz's $H^\infty$ claim fails. A general framework for determining when almost multiplicative operators must be composition operators is developed, leveraging algebraic-consistency, density of polynomials, and bounded evaluation functionals. The paper also completely characterizes Duhamel-multiplicative composition operators on Duhamel algebras, showing they occur precisely for linear self-maps $\varphi(z)=a z$ with $|a|\le1$, and discusses the implications for spaces like the Bloch and little Bloch spaces.

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.

Theorems & Definitions (49)

• Remark 1.1
• Theorem 2.1
• proof
• Remark 2.2
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5
• proof