A covariance equation

TL;DR

This paper characterizes when a covariance-like equation holds for functions on a semigroup, linking measure support to the zero set of a function, with applications to classical semigroups and probabilistic interpretations.

Contribution

It provides a necessary and sufficient condition for a covariance equation involving measures and functions on semigroups, generalizing previous results and solving a posed problem.

Findings

Characterization of measure support in covariance equations

Reduction to the semigroup of non-negative integers

Applications to probabilistic and extremal properties

Abstract

Let $\S$ be a commutative semigroup with identity $e$ and let $\Gamma$ be a compact subset in the pointwise convergence topology of the space $\S'$ of all non-zero multiplicative functions on $\S.$ Given a continuous function $F: \Gamma \to \mathbb C$ and a complex regular Borel measure $\mu$ on $\Gamma$ such that $\mu(\Gamma) \not = 0.$ It is shown that $$ \mu(\Gamma) \int_{\Gamma} \varrho(s) \overline{\varrho(t)} |F|^2(\varrho) d\mu(\varrho) = \int_{\Gamma} \varrho(s) F(\varrho) d\mu(\varrho) \int_{\Gamma} \overline{\varrho(t) F(\varrho)} d\mu(\varrho) $$ for all $(s, t) \in \S\times \S$ if and only if for some $\gamma \in \Gamma, $ the support of $\mu$ is contained is contained in $\{ F = 0 \} \cup \{\gamma\}$. Several applications of this characterization are derived. In particular, the reduction of our theorem to the semigroup of non-negative integers $(\mathbb N_{0}, +)$ solves a problem posed by El Fallah, Klaja, Kellay, Mashregui and Ransford in a more general context. More consequences of our results are given, some of them illustrate the probabilistic flavor behind the problem studied herein and others establish extremal properties of analytic kernels.