A covariance equation

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 ${\S }^{\prime }$ 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 $\mu (\Gamma )\ne 0.$ 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.