A Covariance Equation
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A Covariance Equation El Hassan Youssfi1 Received: 30 November 2018 © Mathematica Josephina, Inc. 2019
Abstract Let S be a commutative semigroup with identity e and let Γ 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 : Γ → C and a complex regular Borel measure μ on Γ such that μ(Γ ) = 0. It is shown that (s)(t)|F|2 ()dμ() = (s)F()dμ() (t)F()dμ() μ(Γ ) Γ
Γ
Γ
for all (s, t) ∈ S × S if and only if for some γ ∈ Γ , the support of μ is contained in {F = 0} ∪ {γ }. Several applications of this characterization are derived. In particular, the reduction of our theorem to the semigroup of non-negative integers (N0 , +) 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. Keywords Toeplitz operator · Finite rank · Covariance equation · Generalized Laplace transforms · Harmonic analysis on semigroups · Bergman kernel Mathematics Subject Classification 47B35 · 30H20 · 43A35 · 43A05 · 46H35
1 Introduction and Statement of the Main Results Let S be a commutative semigroup with identity e, and let S be the completely regular space consisting of all non-zero multiplicative complex-valued functions on S
In memory of Peter Maserick
B 1
El Hassan Youssfi [email protected] Aix-Marseille Université, I2M UMR CNRS 7373, 39 Rue F-Juliot-Curie, 13453 Marseille Cedex 13, France
123
E. H. Youssfi
furnished with the pointwise convergence topology. In this paper, we characterize the compactly supported complex regular Borel measures μ on S that satisfy the following equation: μ(S )
S
(s)(t)|F()|2 dμ() =
S
(s)F()dμ()
S
(t)F()dμ()
for all s, t ∈ S, where F is a fixed continuous function on S . If μ(S ) = 1, then setting χs () := (s), the latter equality can be written in the covariance equation form Fχs − Fχs dμ Fχt − Fχt dμ dμ = 0 S
S
S
for all s, t ∈ S. To handle this problem, we first use the action of the algebra of shift operators on functions on S × S. Then we appeal to Luecking’s Theorem on finite-rank Toeplitz operators [5]. Several applications of our solution will be given in Sects. 5 and 6. They all offer new results. The reduction of our characterization to the additive semigroup N0 of all non-negative integers solves a problem which was left open in recent paper by El Fallah, Kallaj, Kellay, Mashregui, and Ransford in a more general context. Indeed, we provide a solution to this question in the multi-dimensional setting. Let S denote a multiplicative commutative semigroup with identity e. A function : S → C is said to be multiplicative if its satisfies (st) = (s)(t) for all (s, t) ∈ S2 . It is clear, that if is non-zero multiplicative on S, then (e) = 1. We denote by S the set of all non-zero m
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