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.