A COVARIANCE EQUATION

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A COVARIANCE EQUATION

El Hassan Youssfi

To cite this version:

El Hassan Youssfi. A COVARIANCE EQUATION. 2018. �hal-01887296�

EL HASSAN YOUSSFI

ABSTRACT. LetSbe a commutative semigroup with identityeand letΓbe a compact subset in the pointwise convergence topology of the spaceS⁰of all non-zero multiplicative functions onS. Given a continuous functionF : Γ→Cand a complex regular Borel measureµonΓsuch that µ(Γ)6= 0.It is shown that

µ(Γ) Z

Γ

%(s)%(t)|F|²(%)dµ(%) = Z

Γ

%(s)F(%)dµ(%) Z

Γ

%(t)F(%)dµ(%)

for all(s, t)∈S×Sif and only if for someγ∈Γ,the support ofµis contained is contained in {F = 0} ∪ {γ}. Several applications of this characterization are derived. In particular, the reduc- tion 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 results are given, some of them illustrate the probabilistic flavor behind the problem studied herein and others establish extremal properties of analytic kernels.

1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS

LetSbe a commutative semigroup with identityeand letS⁰ be the completely regular space consisting of all non-zero multiplicative complex valued functions onSfurnished with the point- wise convergence topology. In this paper, we characterize the compactly supported complex regular Borel measuresµonS⁰ that satisfy the following equation

µ(S⁰) Z

S⁰

%(s)%(t)|F(%)|²dµ(%) = Z

S⁰

%(s)F(%)dµ(%) Z

S⁰

%(t)F(%)dµ(%)

for all s, t ∈ S, where F is a fixed continuous function on S⁰. If µ(S⁰) = 1, then setting χ_s(%) :=%(s), the latter equality can be written in thecovariance equationform

Z

S⁰

F χ_s−

Z

S⁰

F χ_sdµ F χ_t− Z

S⁰

F χ_tdµ

dµ= 0 for alls, t∈S.

To handle this problem we first use the action of the algebra of shift operators on functions onS×S.Then we appeal to Leucking’s theorem on finite rank Toeplitz operators [4]. Several applications of our solution will be given in Sections 5 and 6. They all offer new results. The reduction of our charaterization to the additive semigroupN0of all non-negative integers solves a problem which was left open in recent paper by El Fallah, Kallaj, Kellay, Mashregui and

Key words and phrases. Toeplitz operator, finite rank, covariance equation, generalized Laplace transforms, harmonic analysis on semigroups, Bergman kernel.

In memory of Peter Maserick.

1

Ransford in a more general context. Indeed, we provide a solution to this question in the multi- dimensional setting.

LetSdenote a multiplicative commutative semigroup with identitye. 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 S⁰ the set of all non-zero multiplicative complex valued functions onSand equip it with pointwise convergence topology under which it is a completely regular topological space.

Let µ be a compactly supported complex regular Borel measure µ on S⁰. It is known (see Rudin [9]) thatµhas a polar decompositionµ= h|µ|, where|µ|is a nonnegative measure, the total variation ofµ, onS^∗ andh =h_µ :S⁰ →Ca Borel measurable function with|h| ≡1.

The generalized Laplace transform ofµis the function defined onS×Sby µ(s, t) :=b

Z

S⁰

%(s)%(t)dµ(%), (s, t)∈S×S. (1.1)

and the variation ofµbis the function

|µ|(s, t) :=c Z

S⁰

%(s)%(t)d|µ|(%), (s, t)∈S×S.

The above generalized Lapalace transforms can be viewed as BV-functions in the sense of Maserick [6] on the semigroup onS×Swith appropriate involution.

IfF is a continuous function onS⁰, we denote byF µthe measure with densityF with respect toµ.

Definition 1. Given complex Borel measure µ on a compact subset Γ of S⁰ and a continuous function F : Γ → C, we shall say that the measure µ satisfies the covariance equation with respect toF if

µ(Γ)|F[|²µ(s, t) =F µ(s, e)c F µ(e, t)c (1.2)

for alls, t∈S.

It is obvious that ifµ(Γ) = 0,thenµsatisfies (1.2) if and only if either Z

Γ

%(s)F(%)dµ(%) = 0 ∀s∈S (1.3)

or

Z

Γ

%(s)F(%)dµ(%) = 0, ∀s∈S. (1.4)

Our goal herein is to describe the measuresµwhich satisfy (1.2). The main result is the follow- ing:

Main Theorem. Suppose thatµis a complex regular Borel measure with compact supportΓ ⊂ S⁰ such thatµ(Γ)6= 0and letF : Γ →Cbe a continuous function such thatF µ 6= 0. Then the following are equivalent

(1) The measureµsatisfies the covariance equation with respect toF.

(2) The measure|µ|satisfies the covariance equation with respect toF.

(3) There are a constantc∈C\ {0}and an elementγ ∈Γsuch thatF(γ)6= 0and µ:=cδ_γ,

(1.5)

whereδγis the Dirac measure atγ.

Indeed, the constantcand the multiplicative functionγ will be given explicitly in terms ofF andµby the equalities

c:=µ(Γ) and γ(s) := ^{F µ(s,e)c}

F µ(e,e)c , s∈S. (1.6)

As a corollary of this, we obtain the following:

Corollary 1. Let µ be a compactly supported complex regular Borel measure on S⁰ such that µ(S⁰)6= 0, and let

γ(s) := µ(s, e)b

µ(S⁰), s∈S. Thenµsatisfies the equality

µ(S⁰) Z

S⁰

%(s)%(t)dµ(%) = Z

S⁰

%(s)dµ(%) Z

S⁰

%(t)dµ(%) for alls, t∈Sif and only ifγ ∈S⁰ andµ=µ(S⁰)δγ.

As a consequence of the latter result we shall derive the following:

Corollary 2. LetY be an integrable complexe random variable such that the expectationE[Y] is inC\ {0}.Then a bounded complex random vectorX = (X₁,· · · , X_d)satisfies the equation

E[Y]E[X^mXⁿY] =E[X^mY]E[XⁿY] (1.7)

for all multi-indices m, n ∈ N^d0 if and only X is constant almost surely. Here, it is understood that

X^m :=X₁^m1· · ·X_d^md for allm= (m1,· · · , md)∈N^d0

The semigroup behind the latter corollary isS= (N^d0,+)introduced in Section 5. Variants of this corollary can be stated and proved for other semigroups, but we will omit to do it.

2. MASERICK’S APPROACH TOBV-FUNCTIONS

Throughout the sequel, (S, e,∗) will denote a multiplicative commutative semigroup with identity e and involution ∗ and consider the associated product semigroup S ×S and furnish it with the involution denoted again by∗and given by

(s, t)^∗ := (t^∗, s^∗), for all(s, t)∈S×S. For each(a, b)∈S×S,we define the shift operatorE_(a,b)by

(E_(a,b)f)(s, t) =f(as, bt)

for all(s, t)∈S×Sandf ∈ C^S×S.The complex spanA(S)of all such operators is a commu- tative algebra with identityE_(e,e)=I and involution



 X

(a,b)

α_(a,b)E_(a,b)





∗

=X

(a,b)

α_(a,b)E_(b,a)

(α_(a,b) ∈ C,(a, b) ∈ S×S). The semigroup S×S is embedded in A(S) as a Hamel basis.

The algebra constructed in this way is isomorphic to theL¹-algebra (or semigroup algebra) con- structed by Hewitt-Zuckermann [9]. The function space C^S×S can be algebraically identified with the dualA(S)^?ofA(S)viaT →(T f)(e, e) (f ∈ C^S×SandT ∈A(S)).This map topologi- cally identifiesC^S×Sequipped with the topology of simple convergence andA(S)^?equipped with the weak*-topology. Following Berg et al. [2],(S×S)^∗ we will denote the set of all semicharac- ters onS×S.That is,(S×S)^∗consists of the set of all membersηof C^S×Ssuch thatη(e, e) = 1 and

η(ss⁰, tt⁰)^∗) =η(s, t)η(s⁰, t⁰) for all(s, t),(s⁰, t⁰)∈S×S.

Equipped with the topology of pointwise convergence,(S×S)^∗is completely regular. It is clear that if%∈S⁰,then

η_%(s, t) :=%(s)%(t), (s, t)∈S×S

is a semicharacter onS×S. Moreover, the mapping% 7→η_%is a homeomorphism fromS⁰ onto (S×S)^∗.This identifies Radon measures onS⁰ with those on(S×S)^∗.

Definition 2. Letτ be a subset ofA(S)and letPτ be convex cone in A(S)spanned by positive linear sums of finite products of elements of τ.The subset τ is called admissible the sense of Maserick if

(1) T^∗ =T for eachT ∈τ.

(2) I−T ∈Pτ for eachT ∈τ.

(3) A(S)is spanned by linear sums of finite products of members ofτ.

An example of an admissible set isτ :={T_a,σ, a,∈S×S, σ∈ {−1,1,−i, i}}where T_a,σ := 1

4

I+σ

2E_a+ ¯σ 2E_a^∗

. (2.1)

Consider a functionf :S×S→Cand letΩbe a collection of subsets ofPτ that satisfies X

T∈Λ

T =I, for all Λ ∈Ω,

{T T⁰ : (T, T⁰)∈Λ×Λ} ∈Ω for all Λ∈Ω

and each element ofτ belongs to someΛ∈Pτ.We impose the ordering onPτ given by Λ1 ≤Λ2 ⇐⇒Λ2 = Λ3Λ2

for someΛ₃ ∈Pτ. Then define

kfk_Λ:=X

T∈Λ

|T f(e, e)|.

It is clear that the functionΛ7→ kfk_Λis nondecreasing. The functionf is said to be of bounded variation (or just aBV-function ) in the sense of Maserick if

kfk:= lim

Λ kfk_Λ<+∞.

We point out that this definition was given by Marserick for arbitrary commutative semigroups with involution. Moreover, in a series of articles [5], [6], [7], he proved thatBV-functions are precisely generalized Laplace transforms of compactly supported complex regular Borel mea- sures on the the space of semi-characters.

To give another way of describing generalized Laplace transforms we recall [2] that a function f :S×S→Cis said to be positive definite if

n

X

j,k=1

c_jc_kf(a_j +a^∗_k) ≥0

forn ∈ Nand finite sequences(c_j) ∈ Cⁿ,and(a_j) ∈ (S×S)ⁿ.It was shown [5], [6], [7] that the space of bounded BV-functions is precisely the complex span of bounded positive definite functions. More generally, it was proved there thatBV-functions complex linear sums of positive definite functions which are bounded in a certain sense (exponential boundedness).

Finally, using the above, Corollary 1 can be rephrased as

Corollary 1’ . Letf :S×S→Cbe aBV-function in the sense of Maserick such thatf(e, e)6=

0.Thenf satisfies the equality

f(e, e)f(s, t) =f(s, e)f(e, t), for all (s, t)∈S×S

if and only if there are a constantc∈C\ {0}and a multiplicative functionγ ∈S⁰ such that f(s, t) =cγ(s)γ(t), for all (s, t)∈S×S.

We point out that in the case of the semigroupS := (]0,1],∧)wheres∧t := min(s, t), with the choice ofτ as in (2.1) thenBV-function on]0,1]×]0,1]in the sense of Maserick coincide withBV-functions in the classical sense. This can be deduced from [6].

3. FINITE RANKTOEPLITZ OPERATORS ON THE UNIT DISC

We recall the basic scheme of Leucking’s method [4] about finite rank Toeplitz operators. Let D := {z ∈ C : |z| < 1}be the unit disc ofC and letA²(D)be classical Bergman space ofD. This is the space of all analytic functions onD which are square integrable with respect to the Lebesgue measuredAonD.The corresponding Bergman kernel is given by

K_D(z, w) := 1

π(1−zw)¯ ², z, w ∈D.

Given a finite complex Borel measureν onD, the Toeplitz operator with symbolν is defined on the spaceC[z]of all analytic polynomials by

T_νf(z) :=

Z

D

f(w)K_D(z, w)dν(w), z ∈D, f ∈C[z].

The measureµinduces a sesquilinear form definedC[z]×C[z], by B_µ(P, Q) :=

Z

C

P(z)Q(z)dν(z)

for allP, Q∈C[z].The reproducing formula for the Bergman kernel shows that B_µ(P, Q) =

Z

D

(T_νP)(z) ¯Q(z)dA(z)

for all analytic polynomials P and Q. We are interested in those bounded Toeplitz operator which onA²(D)which are of finite rank. The characterization of those measures which induce such operators is given by the following result due to Leucking [4].

Leucking’s Theorem . Suppose thatνis a finite complex Borel measure onD. Then for follow- ing are equivalent:

(1) T_ν has finite rank (2) ν has finite support.

Moreover, in the affirmative case, the rank ofT_ν is equal to the number of points in the support ofν.

The latter theorem will be of particular use in the proof of our main theorem.

4. PROOF OF THE MAIN RESULTS

Throughout this section, letΓbe a compact subset ofS⁰. Then for eachs∈Sthe number

|s|Γ:= sup

%∈Γ

|%(s)|

is well-defined and finite due to the continuity of the function%→%(s).

Finally, we point out that ifµsatisfies the covariance equation with respect toF, then in view of (1.2) we have that

Z

Γ

P(%(s))Q(%(t))|F(%)|²dµ(%) = R

ΓP(%(s))F(%)dµ(%)R

ΓQ(%(t))F(%)dµ(%) (4.1)

for alls, t∈Sand all analytic polynomialsP, Q.

Now, fixs∈Sand consider the complex Borel measureν_s onDdefined by Z

D

f(z)dν_s(z) :=R

Γf

%(s) 2(1+|s|_Γ)

|F(%)|²dµ(%) (4.2)

for all compactly supported continuous functionsf onD.

Its obvious that the measureν_sis supported in the closed disc ofCcentered at the origin with radius1/2.

Lemma 4.1. Ifµsatisfies the covariance equation with respect toF, then the Toeplitz operator T_νs induced byν_sis of rank at most1.

Proof. Due to (4.1), it follows that for allP, Q∈C[z]

Z

Γ

P(%(s))|F(%)|²dµ(%) = Z

Γ

P(%(s))F(%)dµ(%) Z

Γ

F(%)dµ(%) Z

Γ

Q(%(s))|F(%)|²dµ(%) = Z

Γ

F(%)dµ(%) Z

Γ

Q(%(s)F(%)dµ(%).

Using this, a little computing shows that for all bounded analytic functionsf onDwe have (T_νsf)(z) =

Z

D

f(w)K_D(z, w)dν_s(w)

= 1 π

+∞

X

j=0

(j + 1)z^j Z

D

f(w)w^jdν_s(w)

= 1 π

+∞

X

j=0

(j + 1)z^j Z

Γ

f

%(s) 2 +|s|_Γ

%(s) 2(1 +|s|_Γ)

!j

|F(%)|²dµ(%)

= 1 π

Z

Γ

f

%(s) 2(1 +|s|_Γ)

F(%)dµ(%)

+∞

X

j=0

(j + 1) Z

Γ

z%(s) 2(1 +|s|_Γ)

!j

F(%)dµ(%)

= Z

Γ

f

%(s) 2(1 +|s|Γ)

F(%)dµ(%) Z

Γ

K_D

z, %(s) 2(1 +|s|Γ)

F(%)dµ(%).

HenceT_νsf belongs to the linear span of the element z 7→

Z

Γ

K_D

z, %(s) 2(1 +|s|_Γ)

F(%)dµ(%) ofA²(D).In particular, ifν_s(D)6= 0,then

(Tνsf)(z) = Z

Γ

f

%(s) 2(1 +|s|_Γ)

F(%)dµ(%) Z

Γ

K_D

z, %(s) 2(1 +|s|_Γ)

F(%)dµ(%)

= R

Γf

%(s) 2(1+|s|Γ)

F(%)dµ(%) R

ΓF(%)dµ(%)

Z

D

K_D(z, w)dνs(w)

= R

Df(w)dν_s(w) ν_s(D)

Z

D

K_D(z, w)dνs(w) This completes the proof.

Lemma 4.2. Suppose thatµsatisfies the covariance equation with respect toF. Then the

Then the following are equivalent.

(1) For alls∈S, the Toeplitz operatorT_νs induced byν_s is of rank1.

(2) There existss∈Ssuch that Toeplitz operatorT_νs induced byν_sis of rank1.

(3) F µ 6= 0.

(4) R

ΓF(%)dµ(%)6= 0.

(5) R

ΓF(%)dµ(%)6= 0.

(6) R

ΓF(%)dµ(%)6= 0andR

ΓF(%)dµ(%)6= 0.

(7) R

Γ|F(%)|²dµ(%)6= 0.

(8) For all s ∈ S, we have R

Γ|F(%)|²dµ(%) 6= 0, and there is a unique complex number a_s ∈Dsuch that

ν_s :=R

Γ|F(%)|²dµ(%)δ_as (4.3)

whereδ_as is the Dirac mass ata_s.

Proof. The facts that(1) ⇒(2),(6)⇒(4)⇒(3),(6)⇒(5)⇒(3)and(8) ⇒(7)are obvious.

That (7) ⇒ (6) follows from (4.1) The proof of Lemma 4.1 shows that (2) ⇒ (3). Finally, assume that (1) holds. By Leucking’s theorem there are complex numbers m_s ∈ C\ {0} and as ∈Dsuch that

ν_s =m_sδ_as. Therefore,

Z

Γ

|F(%)|²dµ(%) = νs(D) =ms6= 0

showing that(8)holds. This completes the proof of the lemma.

We are ready now to prove the Main Theorem.

Proof of Main Theorem. First we prove that (1) ⇒ (3). Assume that µ and F satisfy the hypothesis(1) of the Main Theorem. Without loss of generality we may assume thatµ(Γ) = 1.

Ifµsatisfies the covariance equation with respect toF, then by Lemma 4.2 we see that Z

D

f(z)dν_s(z) = f(a_s) Z

Γ

|F(%)|²dµ(%), for alls∈Sand continuous functions onD). Setting

γ(s) := 2(1 +|s|_Γ)a_s, s∈S, this yields

Z

Γ

f(%(s))|F|²(%)dµ(%) = f(γ(s)) Z

Γ

|F(%)|²dµ(%),

for alls∈Sand continuous functions onC. Takingf(z) =z and applying (4.1) yields γ(s) =

R

Γ%(s)|F|²(%)dµ(%) R

Γ|F(%)|²dµ(%)

= R

Γ%(s)F(%)dµ(z) R

ΓF(%)dµ(%) for alls∈S.Takingf(z) = ¯zand applying (4.1) yields

γ(s) = R

Γ%(s)|F|²(%)dµ(%) R

Γ|F(%)|²dµ(%)

= R

Γ%(s)F(%)dµ(z) R

ΓF(%)dµ(%)

for alls∈S.Using this and applying again (4.1) a little computing shows that for all polynomials P, Q∈C[z]and pairs(s, t)∈S×S,we have

Z

Γ

P(%(s))Q(%(t))|F|²(%)dµ(%) =P(γ(s))Q(γ(t)) Z

Γ

|F(%)|²dµ(%), Now by the Stone-Weierstrass theorem this implies that

Z

Γ

f(%)|F|²(%)dµ(%) = f(γ) Z

Γ

|F(%)|²dµ(%), for all continuous functionsf onΓ∪ {γ}.Henceγ ∈Γand

|F|²µ= Z

Γ

|F|²(%)dµ(%)

δ_γ.

This proves that part (3) of the theorem holds. The proof of the converse if straightforward.

Finally, it is obvious to see that the fact(3) ⇒(2)⇒(1)is true, which completes the proof.

Next we prove Corollary 1.

Proof. Follows by taking the constant functionF ≡ 1in the main theorem. This completes the

proof.

The proof of Corollary 2 will be shifted to the next section.

5. APPLICATIONS TO CLASSICAL SEMIGROUPS

Although the theory works for general semigroups, we limit ourselves concrete semigroups to illustrate the results. There are many interesting examples of such semigroups for which the multiplicative functions are explicit [2].

5.1. The semigroup (N^d0,+). For a positive integer d the semi-group (N^d0,+) is the set of all multi-indicesm= (m₁,· · · , m_d), where the entries are nonnegative integers, equipped the stan- dard addition. The cased = 1corresponds to the setting of the question posed in the paper [3]

asking for the description of those signed measures µon the closed unit discDthat satisfy the equality

µ(D) Z

D

zⁿz¯^mdµ(z) = Z

D

zⁿdµ(z) Z

D

¯

z^mdµ(z) (5.1)

for all m, nin the set N0 of all non-negative integers. A direct application of our main result solves this problem even in a more general abstract setting. Indeed,

Proposition 1. Letdbe a positive integer. Suppose thatF is continuous function onC^dandµis a compactly supported complex Borel measure onC^dsuch that Ifµ(C^d) 6= 0andF µ 6= 0,then the following are equivalent

(1) µsatisfies the covariance equation with respect toF. µ(C^d)

Z

C^d

zⁿz¯^m|F(z)|²dµ(z) = Z

C^d

zⁿF(z)dµ(z) Z

C^d

¯

z^mF(z)dµ(z) (5.2)

for all multi-indicesm, n∈N^d0

(2) There are a constant c ∈ C\ {0} and a vector ζ ∈ C^d such that F(ζ) 6= 0such that µ=cδ_ζ.

Proof. As mentioned before, we consider the finitely generated abelian semigroupS = (N^d0,+) with neutral element 0. It is not hard to see the corresponding semigroupS⁰ can be identified algebraically and topologically withC^dfollowing the correspondence

%_z(m) =z^m, m∈N^d0, z∈C^d

with the understanding that0⁰ = 1.To complete the proof, it suffices to apply Corollary 1 to this

semigroup.

Remark 1. Under the assumptions of Proposition 1, if F ≡ 1andµ(C^d) 6= 0,thenµsatisfies (5.2) if and only if there is a nonzero constantc∈Csuch thatµ=cδ_ζ where

ζ := 1 µ(C^d)

Z

C^d

z₁dµ(z),· · · , Z

C^d

z_ddµ(z)

.

Hence the reduction of this result to the cased = 1and F ≡ 1solves the problem (5.1) posed by [3].

Now we prove Corollary 2.

Proof of Corollary 2. Suppose thatY is an integrable random variable with expectationE[Y]6=

0.LetX be a bounded complex random vector and which satisfies (1.7). We may assume that the complex random vectorXtakes its values inside a compactΓofC^d. Consider the compactly supported complex BorelµonC^ddefined by

Z

C^d

f(z)dµ(z) =E[f(X)Y] (5.3)

for all continuous function onC^d supported inΓ. Thenµ(C^d) = E[Y]6= 0andµsatisfies (5.2) withF ≡1so that by Proposition 1 we see that for some vectorζ ∈C^dand a nonzero constant cwe haveµ=cδ_ζ.Hence

E[f(X)Y] =cf(ζ)

for all bounded Borel measurable functions f on C^d. Denote by E[Y|X = x] the conditional expectation ofY givenX =z. Then the latter equality implies that

Z

C^d

E(Y|X =z)f(x)dP_X(z) =cf(ζ)

for all bounded Borel measurable functionsf onC^dshowing that dP_X =δ_ζ and E(Y|X =ζ) = c=E(Y).

The converse is obvious, completing the proof.

5.2. The multiplicative semi-group(N, .). This is the set of all positive integers equipped the natural product with neutral element1.Let(p_j)_j∈Nbe the sequence of primes. The fundamental theorem of arithmetic ensures that eachn ∈ Nadmites a unique representationn = pⁿ₁¹· · ·pⁿ_l^l, wherel∈Nandn₁· · ·n_l ∈N0.we setκ(n) := (n₁,· · · , n_l)and for each sequencez = (z_j)j∈N, of complex numbers, we write

z^κ(n):= Π^l_j=1z_j^nj.

It is not hard to see that the topological semigroup(N, .)⁰ of multiplicative functions on(N, .) can be identified algebraically and topologically with C^∞ := C^N be furnished with product topology following the correspondence

%_z(n) = z^κ(n), n∈N, z ∈C^∞. The reduction of our main result to this semigroup gives the following.

Proposition 2. LetF : C^∞ → Cbe a continuous function and letµbe a compactly supported complex regular Borel measure onC^∞ such thatµ(C^∞) 6= 0andF µ 6= 0.Thenµsatisfies the covariance equaltion

µ(C^∞) Z

C^∞

z^κ(n)z¯^κ(m)|F(z)|²dµ(z) = Z

C^∞

F(z)z^κ(n)dµ(z) Z

C^∞

z^κ(m)F(z)dµ(z) (5.4)

for all multi-indicesm, n ∈ Nif and only if there are a nonzero constantcand an elementζ of C^∞such thatF(ζ)6= 0andµ=cδ_ζ.

Remark 2. Under the assumptions of Proposition 2, ifF is the constant function1andµ(C^∞)6=

0,then thenµsatisfies (5.4) if and only if there is a constantc∈Csuch thatµ=cδ_ζ where ζ := 1

µ(C^∞) Z

C^∞

z₁dµ(z),· · · , Z

C^∞

z_ddµ(z)

.

5.3. The semigroup S = ([0,+∞),+). The continuous multiplicative on this semigroup are form%_z(s) =e^sz, z ∈C. Consider the subset ofS⁰ given by

Γ :={s 7→e^−sz, z∈C, Rez ≥0} ∪ {∞}

with the understanding that%∞(s) = 1{0}(s), s ≥0.ThenΓis a compact subset ofS⁰.Further- more, complex regular Borel measures onΓ\ {∞}are in a bijective correspondence with finite complex regular measure on the right halfplane H = {z ∈ C, Rez ≥ 0}.Hence every finite complex regular measure µon H induces a compactly supported finite complex regular Borel measureν onΓdefined by its generalized Laplace transform

bν(s, t) = Z

Γ\{∞}

%(s)%(t)dν(%) = Z

H

e^−sz−t¯zdµ(z), (s, t)S×S. Applying the main theorem to this semigroup yields the following

Proposition 3. Let F : H → C be a continuous function and letµ be a finite complex Borel measure onH such thatµ(H) 6= 0andF µ 6= 0.Thenµsatisfies the covariance equation with respect toF

µ(H) Z

H

e^−sz−t¯z|F(z)|²dµ(z) = Z

H

e^−szF(z)dµ(z) Z

H

e^−t¯zF(z)dµ(z) (5.5)

for alls, s ∈ [0,+∞)if and only if there are a nonzero constant cand an elementζ ofHsuch thatF(ζ)6= 0andµ=cδζ.

Remark 3. The semigroup S = ([0,+∞),+) is a sample example of the so called conelike semigroups. These are subsetsSof finite dimensional vector spaces that are stable under addition, contain the origin0and satisfy: for allx∈S,there is a real numberr_x ≥0such thatrx∈Sfor allr ≥ r_x.They were introduced by Ressel [8] in connection with Bochner’s type theorem for topological semigroups. Using the work [10], these semigroups were shown to be of interest to holomorphic functions in tube domains [11]. Finally, our main theorem can be applied to derive results in the same spirit as Proposition 3 in the case of these semigroups.

6. ANALYTIC KERNELS

This section offers further applications of our results. We will establish a kind of extremal property for analytic kernels. For simplicity we consider the ball setting. Consider and open B^d in C^d and a closed ballB^p inC^p both centered at 0with positive radius. Assume that K : Bd×Bp →Cbe kernel of the form

K(z, w) = X

m,n∈N^d0

a_m,nz^mwⁿ, (6.1)

with nonzero complexe coefficientsam,n, where the series is uniformly convergent in the variable wfor all fixedz.We will prove the following

Theorem 1. Suppose thatK is a kernel of the form (6.1) and letf : Bd → C be holomorphic function. Then a finite complex Borel measureµonB^psuch thatµ(B^p) = 1satisfies the equality

|f(z)|² = Z

Bp

|K(z, w)|²dµ(w), (6.2)

in a neighborhood of0if and only if there is are complex constantcand an elementζofB^p such that

µ=cδ_ζ and f =cK(.,ζ)¯ Proof. We expandf in terms of a power series f(z) =P

m∈N^d0 b_mz^m in a neighborhood of0. If µsatisfies (6.2), then by integration, uniqueness of the power series expansion and (6.1) we see that

b_pb_q =a_m,qa_p,n Z

Bp

wⁿw^mdµ(w),

for allm, n, p, q∈N^d0.The proof now follows from Proposition 1.

7. CONCLUDING REMARKS

Remark 4. In view of Lemma 4.1 in [10] we see that generalized Laplace transforms of non- negative measure are preserved under surjective morphisms of semigroups. This, combined with Marserick’s work [6], shows that generalized Laplace transforms of compactly supported com- plex regular Borel measures remain preserved under these mappings. Below, we list a couple examples:

Since the mappingx 7→ e^x is an isomorphism from the additiveS = ([0,+∞),+) onto the multiplicative semigroupT = ([1,+∞), .)The analog of Proposition 3 on Tcan be derived by a simple change of variable.

Every finitely generated abelian semigroup Twith unit is surjective image of (N^d0,+,0) for somed∈N,hence a similar observation shows an analog of Proposition 1 onTcan be deduced i by change of variable.

REFERENCES

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100, Springer-Verlag, Berlin/New York, 1984

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[4] D. Leucking,Finite rank Toeplitz operators on the Bergman spaceProc. AMS136(2008), 1717–1723 [5] P. H. Maserick,Moments of measures on convex bodies, Pacific J. Math.68(1977), 135-152.

[6] P. H. Maserick,BV-functions, positive-definite functions and moment problems, Trans. Amer. Math. Soc.214 (975, 137-152

[7] P. H. Maserick,Moment and BV-functions on commutative semigroups, Trans. Amer. Math. Soc.181(1973), 61-75 .

[8] P. Ressel,Bochner’s theorem for finite-dimensional conelike semigroups, Math. Ann.296(1993), no. 3, 431- 440.

[9] W. Rudin,Real and complex analysis,McGraw-Hill, New York, 1966.

[10] E. H. Youssfi, Pull-back properties of moment functions and a generalization of Bochner-Weill’s theorem, Math. Ann.300(1994), 435-450.

[11] E. H. Youssfi,Harmonic analysis on conelike bodies and holomorphic functions on tube domains,Journal of Functional analysis.155(1998), 381-435

E-mail address, E.H. Youssfi:el-hassan.youssfi@univ-amu.fr