A Covariance Equation

Abstract

Let $\mathbb {S}$ be a commutative semigroup with identity e and let $\varGamma$ be a compact subset in the pointwise convergence topology of the space $\mathbb {S}'$ of all non-zero multiplicative functions on $\mathbb {S}.$ Given a continuous function $F: \varGamma \rightarrow \mathbb {C}$ and a complex regular Borel measure $\mu$ on $\varGamma$ such that $\mu (\varGamma ) \not = 0.$ It is shown that

$$\begin{aligned} \mu (\varGamma ) \int _{\varGamma } \varrho (s) \overline{\varrho (t)} |F|^2(\varrho ) \mathrm{d}\mu (\varrho ) = \int _{\varGamma } \varrho (s) F(\varrho ) \mathrm{d}\mu (\varrho ) \int _{\varGamma } \overline{\varrho (t) F(\varrho )} \mathrm{d}\mu (\varrho ) \end{aligned}$$

for all $(s, t) \in \mathbb {S}\times \mathbb {S}$ if and only if for some $\gamma \in \varGamma ,$ the support of $\mu$ 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.