Hankel operators and the Stieltjes moment problem

Transcription

1 Journal of Functional Analysis 58 1) Hankel operators an the Stieltjes moment problem Hélène Bommier-Hato, El Hassan Youssfi LATP, U.M.R. C.N.R.S. 663, CMI, Université e Provence, 39 Rue F-Joliot-Curie, Marseille Ceex 13, France Receive 19 June 9; accepte 9 August 9 Available online 1 September 9 Communicate by N. Kalton Abstract Let s be a non-vanishing Stieltjes moment sequence an let μ be a representing measure of it. We enote by μ n the image measure in C n of μ σ n uner the map t, ξ) tξ,whereσ n is the rotation invariant probability measure on the unit sphere. We show that the closure of holomorphic polynomials in L μ n ) is a reproucing kernel Hilbert space of analytic functions an escribe various spectral properties of the corresponing Hankel operators with anti-holomorphic symbols. In particular, if n = 1, we prove that there are nontrivial Hilbert Schmit Hankel operators with anti-holomorphic symbols if an only if s is exponentially boune. In this case, the space of symbols of such operators is shown to be the classical Dirichlet space. We mention that the classical weighte Bergman spaces, the Hary space an Fock type spaces fall in this setting. 9 Elsevier Inc. All rights reserve. Keywors: Hankel operator; Fock space; Bergman kernel 1. Introuction In this paper we consier Hankel operators an the -canonical solution operator in a Hilbert space of analytic functions relate to a Stieltjes moment sequence. We recall that a sequence s = ), N, is sai to be a Stieltjes moment sequence if it has the form This research is partially supporte by the French ANR DYNOP, Blanc * Corresponing author. aresses: bommier@gyptis.univ-mrs.fr H. Bommier-Hato), youssfi@gyptis.univ-mrs.fr E.H. Youssfi). -136/$ see front matter 9 Elsevier Inc. All rights reserve. oi:1.116/j.jfa.9.8.4

2 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) = + t μt), where μ is a non-negative measure on [, + [, calle a representing measure for s. These sequences have been characterize by Stieltjes [] in terms of some positive efiniteness conitions. We enote by S the set of such sequences. It follows from the above integral representation that each s S is either non-vanishing, that is, > for all, orelse = δ for all. We enote by S the set of all non-vanishing elements of S. Fix an element s = ) S.By Cauchy Schwarz inequality we see that the sequence +1 is non-ecreasing an hence converges a + to the raius of convergence of the entire series F s λ) := + =n 1 λ +1 n, λ C. Set R s := lim s = lim +. The sequences s for which the raius R s is finite are calle exponentially boune [5]. Denote by the ball in C n centere at the origin with raius R s with the unerstaning that = C n when R s =+. We enote by A s) the Hilbert space of those holomorphic functions fz)= α N n a αz α on that satisfy α N n α!s α α +n 1)! a α < + equippe with the natural inner prouct f,g := α N n α!s α α +n 1)! a α b α if fz)= α N n a αz α an gz) = α N n b αz α are two elements of A s). Now let σ = σ n be the rotation invariant probability measure on the unit sphere S n in C n an let μ be a representing measure of s. We enote by μ n the image measure in C n of μ σ n uner the map t, ξ) tξ from [, + [ S n onto C n. We consier the Hilbert space L μ n ) of square integrable complex-value functions in C n with respect to the measure μ n. Our first result is the following: Theorem A. The measure μ n is supporte by the closure of the omain. In aition, for each set compact K there exists C = CK) > such that sup fz) C f L μ n ) z K

3 98 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) for all holomorphic polynomials f in C n. Furthermore, the space A s) coincies with the closure of the holomorphic polynomials in L μ n ) an its reproucing kernel is given by K s z, w) = 1 n 1)! F s n 1) ) z, w, z,w Ωs. The classical weighte Bergman spaces, weighte Fock spaces an Hary spaces are of the form A s); each of these space is associate to an appropriate choice of the sequence s, see [,,6]. To state further results we consier the orthogonal projection P s associate to A μ n ).Itis given for all g L μ n ) by P s g)z) = K s z, w)gw) μ n w), z. This integral operator can be extene in a natural way to functions g that satisfy K s z, )g L 1 μ n ) for all z. This extension allows us to efine Hankel operators. To o so, enote by Ts) the class of all f A s) such that fϕk s z, ) L 1 μ n ) for all holomorphic polynomials ϕ an z an the function Hf ϕ)z) := C n K s z, w)ϕw) [ fz) fw) ] μ n w), z, is the restriction to of a function in L μ n ). This is a ensely efine operator from A s) into L μ n ) which will be calle the Hankel operator Hf with symbol f. It can be written in the form Hf ϕ) = I P s) fϕ) for all holomorphic polynomials ϕ. It is not har to see that the class Ts) contains all holomorphic polynomials. Finally, if f Ts), we enote by Specf ) the set of all multi-inices k N n such that k f ). z Our secon result is the following k Theorem B. Suppose that f is a holomorphic polynomial. Then 1) Hf is boune if an only if for all k Specf ). ) Hf is compact if an only if for all k Specf ). s+ k sup + k + n 1 N + k s+ k lim + k + n k ) + k < + 1.1) ) + k = 1.)

4 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) ) If p>, then H f is in the Schatten class Sp A s), L μ n )) if an only if N for all k Specf ). n 1 s+ k s ) p + k + k + n 1) n 1 p s+ k ) p < + We point out that if the sequence s is exponentially boune then 1.1) an 1.) hol. The last assertion of Theorem B shows that if n, an the Schatten class S p A s), L μ n )) contains nontrivial Hankel operators with anti-holomorphic symbols, then p>n. The converse to this statement is not true as shown by the authors in [11]. In particular, in higher imensions there are no nontrivial Hilbert Schmit Hankel operators with anti-holomorphic symbols. The situation in the one-imensional case is completely ifferent. More precisely: Theorem C. Suppose that n = 1 an f is a nonconstant holomorphic function in f Ts). Then Hf is in the Hilbert Schmit class S A s), L μ n )) if an only if s is exponentially boune an f is in the classical Dirichlet space D ). In aition, the trace TrH f H f ) of H f H f is given by Tr H f H ) 1 f = π = f z) Az) fz) fw) K s z, w) Az)Aw) where Az) is the Lebesgue measure in C. The first equality shows the characterization in the latter theorem epens only on the limit s lim The above result has been prove by separate methos in the two simple particular cases of Hary an Bergman spaces [6]. Now we shall characterize the bouneness, the compactness an the membership in a Schatten class of S the canonical solution operator of the on the space H,1) ) consisting of, 1)-forms with holomorphic coefficients in L μ n ) efine by Sf) = f an Sf is orthogonal to holomorphic elements of L μ n ). The spectral properties of this operator were stuie by Haslinger [1,13], Haslinger an Helfer [14] an Lovera an Youssfi [17]. Corollary 1.1. Consier the canonical solution operator S to the from H,1) ) to L μ n ). Then the following are equivalent: 1) S is boune on H,1) ). ) For all j = 1,...,n, the Hankel operator H zj is boune from A s) into L μ n ). 3) There is j = 1,...,n, such that the Hankel operator H zj is boune from A s) into L μ n ).

5 98 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) ) There is a positive constant C>such that s+n s ) +n 1 + n 1 +n 1 +n for all positive integer. +n +n 1 C Corollary 1.. Consier the canonical solution operator S to the from H,1) ) to L μ n ). Then the following are equivalent: 1) S is compact on H,1) ). ) For all j = 1,...,n, the Hankel operator H zj is compact from A s) into L μ n ). 3) There is j = 1,...,n, such that the Hankel operator H zj is compact A s) into L μ n ). 4) We have s+n lim +n 1 + n 1 + +n 1 +n +n +n 1 ) =. In each of the two preceing corollaries, the equivalence between the two assertions 1) an 4) was establishe in Lovera an Youssfi [17] an later by Haslinger an Lamel [15]. Corollary 1.3. Consier the canonical solution operator S to the from H,1) ) to L μ n ) an let p>. Then the following are equivalent: 1) S is in the Schatten class S p H,1) )L μ n )). ) For all j = 1,...,n, the Hankel operator H zj is in the Schatten class S p A s), L μ n )). 3) There is j = 1,...,n, such that the Hankel operator H zj is in the Schatten class S p A s), L μ n )). 4) There is a positive constant C such that N for all positive integer. n 1 s+n s ) p ) p +n 1 + n 1) n 1 p s+n C +n 1 +n +n 1 In the latter corollary, the equivalence between the two assertions 1) an 4) was establishe in Lovera an Youssfi [17] in the case p an later by Haslinger an Lamel [15] in the general case. To state another result, we let Ms) be the subspace of Ts) consisting of those functions f for which the Hankel operator H f is boune on A s). We equip Ms) with norm f Ms) := H f + f). The subspace of Ms) consisting of functions f such that Hf is a compact operator will be enote by M s). Then it is not har to see that M s) is a close subspace of Ms).

6 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) If p>, we enote by M p s) the subspace of those functions f Ms) such that the Hankel operator H f is the Schatten class S pa s), L μ n )). We equip M p s) with quasi-norm Then we have the following f Mp s) := H f S p + f). Theorem D. Let X {Ms), M s), M p s)} an let U be a rotation in C n. Then the following assertions hol. 1) If f X, then f U X an f U X = f X. ) If f X, then z k X for all k Specf ). 3) If the sequence s is either exponentially boune or satisfies s+l lim + s ) 1 = for all l N, 1.3) then the spaces Ms), M s) an M p s), p 1, are Banach spaces an the space M p s), <p<1, is a quasi-banach space. We point out that there are examples of Stieltjes moment sequences that o not satisfy 1.3) as shown by Boas type sequences [1]. There is a sequence of positive real numbers s satisfying s 1 an s n+1 ns n ) n+1. It is not har to see by Theorem B that the spaces Ms), M s) an M p s) corresponing to such sequences are trivial, namely, they consist only of constant functions. Another type of Stieltjes moment sequences for which Theorem B applies to show that the corresponing spaces Ms), M s) an M p s) are trivial are the Stieltjes sequences s that satisfy s 1 an s δ+1 1 for all 1 1.4) for some <δ<1. Arbitrary sequences satisfying 1.4) were stuie by Bisgaar an Sasvári [9] an Bisgaar [8]. They were shown in [8] to be Stieltjes moment sequences as long as 1 δ The Hilbert space A s) an relate operators We first fix some notations. Let N n enote the set of all n-tuples with components in the set N of all non-negative integers. If α = α 1,...,α n ) N n,welet α :=α 1 + +α n enote the length of α. Ifβ = β 1,...,β n ) N n satisfies α j β j for all j = 1,...,n, then we write α β. Otherwise, set α β. Finally, if A an B are two quantities, we use the symbol A B whenever A C 1 B an B C A, where C 1 an C are positive constants inepenent of the varying parameters. Proof of Theorem A. We first observe that if a positive real number r satisfies μ]r, + [) =, then for all non-negative integer, wehave r μ], + [) an hence lim sup 1 r.

7 984 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) This shows that the raius of convergence of the series F s is smaller than or equal to the infimum of all such real numbers r. Conversely, suppose that r> satisfies μ]r, + [)>. Then for all non-negative integer. Therefore, Since r μ ]r, + [ ) r lim inf 1 1 lim sup. sup { r: μ ]r, + [ ) > } = inf { r: μ ]r, + [ ) = }, we see that Rs = lim 1 +. Therefore, the measure μ n is supporte by the closure. Since both series F s an F s n 1) have the same raius of convergence it follows that for each z, the series K s z, w) = n 1)! z, w, w, n 1)!! converges on. Moreover, by Fatou s lemma an orthogonality of the holomorphic monomials with respect to μ n we have Ks z, w) ) 1 N μ n w) lim inf n 1)! N + ) 1 = lim inf n 1)! N + = K s z, z). = = + n 1)!! z, w μ n w) N + n 1)! z, w!s μ n w) Hence for any fixe z, the series K s z, w) converges in L μ n ). In aition, a little computing shows that for any α N n,wehave w α n 1)! K s z, w) μ n w) = n 1)!! = 1 α +n 1)! = n 1)! α!s α = z α. w α z, w μ n w) w α z, w α μ n w)

8 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) This shows that the kernel K s z, w) reprouces holomorphic polynomials. Moreover, it satisfies sup fz) sup Ks z, z) f L μ n ) z K z K for all holomorphic polynomials an each set compact K Ω. The remaining part of the proof follows by stanar arguments. We point out that R s is always strictly positive. We recall from the previous work of the authors [11] the expression of the operators H z k an H z k H z l on holomorphic homogeneous polynomials. Lemma.1. Suppose that k an l are in N n. Then the omain DomH z ) of H z contains all k k polynomials in w an w. Moreover, if f is a holomorphic homogeneous polynomial of egree, then H z l H z k f = + l Γn+ + l k ) + l k Γn+ + l ) k z l z k f ) Γ+ n k ) k Γ+ n) z l k z k f. In particular, H z l H z kf is a holomorphic homogeneous polynomial of egree + l k. In particular, for each α in N n, the monomial zα is an eigenvector for the operator H z k H z k an the corresponing eigenvalue λ α is given by if α k an otherwise. λ α = s α + k s α Γn+ α ) α + k)! Γn+ α + k ) α! λ α = s α + k s α s α Γn+ α ) Γn+ α + k ) For simplicity reasons, we introuce some notations. We set with the unerstaning that k j tk t j f n t 1,...,t n ) := k t k + Γ α +n k ) s α k Γ α +n) n k j = as long as k j = an t k j j t j α + k)!, α! α! α k)! t k t j, t R n,.1) = t k j 1 j 1 + α1 tα):= α +n,..., 1 + α ) n, α N n α +n. for k j 1. We also let Lemma.. The function f n given by.1) satisfies f n t 1,...,t n ) for all non-negative real numbers t 1,...,t n that satisfy t 1 + +t n = 1. In particular, f n tα)) for all α N n.

9 986 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Proof. Setting r j = k j k, the lemma follows from the inequality n r j t j 1, which hols for all t 1,...,t n,r 1,...,r n ], + [ that satisfy t 1 + +t n = r 1 + +r n = 1. This inequality, in turn, can be prove by inuction on n. Lemma.3. Suppose that α an k are in N n.ifn = 1,setγ α,k := an if n>1,set γ α,k := 1 Γn+ α ) α + k)! Γ α +n k ) n 1 Γn+ α + k ) α! Γ α +n) Then γ α,k, for all α N n that satisfy α k. In aition, if n, then γ α,k = 1 )) 1 ) 1 f n tα) + O, n 1 + n for all k,α N n, satisfying α k, where := α. α! α k)! Proof. We consier the particular case of the constant Stieltjes moment sequence = 1, N, represente by the Dirac measure μ = δ 1.Ifα N n, then n 1)γ α,k is the eigenvalue of H z k H z k corresponing to the eigenvector z α. Applying the previous lemma we see that n 1)γ α,k an hence the first part of the lemma follows. Next, we prove the secon part of lemma. From the property of the Gamma function [18] )) Γx+ y) Γx+ z) = y z)y + z 1) 1 xy z O x x as x +, ). where y an z are real numbers, we get Γ+ n) = + n) k 1 k k 1) 1 + O Γ+ n + k ) + n) Γ+ n k ) Γ+ n) = + n) k 1 + k k +1) + O + n) 1 )) )) a +, a +. By the proof of Lemma 3.3 in [11], we have, when α k, α + k)! α! = n 1 + α j ) k j + n [ = + n) k t k ht) gt) + + qα) ] + n + n) k k j k j 1) 1 + α j ) k j 1 + α l ) l j1 k l + qα)

10 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) where ht) := n k j tk t j, gt):= n k j t k t j. Using a similar argument, we also have α! n α k)! = 1 + α j ) k j n [ = + n) k t k ht) + gt) + rα) ] + n + n) k where q an r are polynomials of egree at most k. k j k j + 1) 1 + α j ) k j 1 + α l ) l j1 k l + rα) Γ+ n) α + k)! Γ+ n k ) α! Γ+ n + k ) α! Γ+ n) α k)! k α + k)! = + n) 1 k k 1) 1 + O α! + n) + n) k α! 1 + k k +1) α k)! + n) = 1 k k 1) 1 + O + n) 1 + k k +1) 1 + O + n) = 1 k t k + ht) + O + n + O )) 1 ))[ t k ht) gt) + + n )) ))[ t k ht) + gt) + n 1 )). )] 1 + O + O )] 1 The lemma now follows since f n t) = k t k + ht). Lemma.4. If α N n, then the eigenvalue λ α of the operator H z k H z k satisfies s α + k λ α = s ) )) α tα) ) k 1 + O + n 1 s α s α k + n s α s α k )) ) 1 f n tα) + O if α k an λ α = s ) α + k 1 O, s α otherwise.

11 988 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Proof. By Lemma.1 an the efinition of γ α,k,wehave s α + k λ α = s ) α Γ+ n) α + k)! s α + n 1)γ n,k. s α s α k Γ+ n + k ) α! s α k By the estimates in the proof of Lemma.3 we euce that )) Γ+ n) α + k)! 1 = t tα) + O. Γ+ n + k ) α! The latter equation, combine with Lemma.3, completes the proof of the first part of the lemma. To prove the remaining part of the lemma, suppose that for some j = 1,...,n we have that k j 1 an α j <k j. Then by Lemma.1 we have λ α = s α + k s α Γ+ n) Γ+ n + k ) α + k)!. α! Set α = α 1,...,α j 1,,α j +1,...,α n ) an k = k 1,...,k j 1,,k j +1,...,k n ). Arguing like in Lemma 3.4 in [11], we get α + k)! α! [ n n k k j )! 1 + α j ) kj j k j 1) α j ) ] k j α s ) kl,j j,j j l j,j + qα ), where qα ) is a polynomial of egree at most k. This inequality, combine with the estimate ) Γ+ n) Γ+ n + k ) = O 1 + n) k gives the secon part of the lemma. Theorem.5. Fix k N n an consier the Hankel operator H z k A s) consisting of holomorphic polynomials into L μ n ). Then: 1) H z k is boune if an only if ) H z k is compact if an only if s+ k sup + k + n 1 N + k s+ k lim + k + n k from the ense subspace of ) + k < +..) ) + k =..3)

12 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Proof. We consier the sequence λ α ) α of eigenvalues of the H z k H z k.letσ n be the simplex consisting of those t = t 1,...,t n ) R n such that t j an t 1 + +t n = 1. Since the set { α1 + 1 N + n,...,α n n ) }, α = iense in Σ n, it follows that ) α1 + 1 sup f n α = + n,...,α n + 1 sup f n t) + n t Σ n an sup tα) k sup t k α = t Σ n a tens to +. These estimates, combine with Lemma.4, implies that λ α ) α is boune if an only if.) hols an lim α + λ α = if an only if.3) hols. The theorem now follows since H z k is boune if an only if H z k H z k is boune an compactness of H z k is equivalent to that of H z k H z k. Next, let p>. We shall stuy the membership of the operator H z k in a Schatten class S p. Recall that H z k is in S p if an only if H z k H z k is in S p, that is to say the series p λα is convergent. Let be an integer. We shall estimate the sum S = α = λp α, when +. The calculations above lea to stuy the cases α k an its opposite separately. Let B := {α N n, α =}. We partition B = B B, where B ={α B,α k} an B = B \ B. Thus S can be written in the form S = S + S, where S = α B λp α an S = α B λp α.we shall use the following lemmas see [11]). Lemma.6. If n, then we have the estimates B B n 1)! n 1 an B n as +. Lemma.7. Suppose that n an g is a continuous function on R n 1. Consier the open set D := {t 1,...,t n 1 ) R n 1 +, n 1 t j < 1}. For a multi-inex β = β 1,...,β n 1 ) in N n 1,set β1 + 1 c β, := { n 1 J := β N n 1 : Then lim + 1 n 1 β J gc β, ) = D gt)t.,..., β n [ βj, β j + 1 ), ] } D. The above results enable us to estimate S when = α +.

13 99 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Lemma.8. If p>, then S n 1 s+ k s ) p ) p + n 1) n 1 p s. k k Proof. Recall that S = S + S, where S = α B λp α an S = α B λp α. First we shall estimate S. By Lemma.4, we know that this sum has the following expansion when = α + S s+ k s ) p k α B ) p n n k α B )) tα) ) k 1 p + O f n tα) ) + O 1 )) p. Using the properties of the function x x p an Lemma.7 we see that there exists a constant M>, such that )) tα) ) k 1 p + O n 1 α B D )) ) 1 p f n tα) + O n 1 α B Therefore, S n 1 s+ k s α k n 1 s+ k s α k a +. To estimate S we observe that if n = 1, then S S k. D t pk 1 1 t pk n 1 n 1 1 ) pkn n t j t, f n t 1,...,t n 1, 1 ) p n 1 + n 1 + n k ) p + n 1) n 1 p s )) p n t j t. ) p k On the other han, if n, by Lemma.4 we see that for α B we have ) p λ p α = n 1) s+ k O p). Since B n, we see that S = OS + k p ). The lemma follows from the relation S = S + S. We then characterize the Schatten class membership of H z k. ) p

14 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Theorem.9. Let k N n. Then the Hankel operator H z k is L μ n )) if an only if in the Schatten class Sp A s), n 1 s+ k s ) p + k + k + n 1) n 1 p s+ k ) p < +..4) Proof. We use that the operator H z k is in the Schatten class S p A s), L μ n )) if an only if H z k H z k is in S p A s)). Therefore, the theorem follows from Lemma.8. Lemma.1. If U is a unitary transformation in C n, the operator Uf := f U is a unitary isometry from L μ n ) onto itself an from A s) onto itself. Moreover the following assertions hol. 1) If f Ms), then Uf Ms) an Uf Ms) = f Ms). ) If f M s), then Uf M s). 3) If f M p s), then Uf M p s) an Uf Mp s) = f Mp s). Proof. Let U be a unitary transformation in C n an enote U its ajoint, which is also its inverse. It is clear that the operator U is a unitary isometry from L μ n ) onto itself an from A s) onto itself. Let f be in Ms). Ifg is a holomorphic polynomial, then by a change of variable we see that Therefore, H Uf g)z) = K s Uz, w)gu w) C n = K s Uz, w)u g)w) C n = Hf U g)uz) = UHf U )g)z). an thus H Uf = Hf, showing that [ Uf z) fw) ] μ m w) [ fuz) fw) ] μ n w) H Uf = UH f U.5) Uf Ms) = f Ms). This proves part 1) of the lemma. The proof of parts ) an 3) of the lemma are similar. Let T n := {ζ = ζ 1,...,ζ n ) C n : ζ j =1, j= 1,...n} an for ζ = ζ 1,...,ζ n ) T n,let U ζ be the unitary linear transformation in C n efine by U ζ z) = ζ 1 z 1,...,ζ n z n ), for all z = z 1,...,z n ) C n. Like in [1] we have the following

15 99 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Lemma.11. If f Ts) an g A s), the mappings ζ U ζ g an ζ H Uζ f g) are continuous from T n to L μ n ). Proof. Let g A μ n ) an write gz) = α N n a kz α.ifζ,η T n, then U ζ U η )g L μ n ) = g U ζ g U η L μ n ) = a α Uζ z) α U η z) α L μ n ) α N n = a α c α ζ α η α, α N n where Since c α = C n z α μ n z), α N n. a α c α < + an ζ α η α 4, α N n the ominate convergence theorem leas to lim U ζ U η )g L μ n ) =, ζ η showing that the mapping ζ U ζ g is continuous from T n to L μ n ). This, combine with the fact that U ζ is unitary an the equalities H Uζ f H U η f = U ζ H f U ζ U ηh f U η = U ζ H f U ζ U ζ H f U η + U ζ H f U η U η H f U η = U ζ Hf U ζ U η) + U ζ U η )Hf U η, shows that the mapping ζ H Uζ f g) is also continuous from Tn to L μ n ). Lemma.1. Assume that f Ts). 1) If f Ms), then for any multi-inex k Specf ), the monomial z k is in Ms). ) If f M s), then for any multi-inex k Specf ), the monomial z k is in M s). 3) If p> an f M p s), then for any multi-inex k Specf ), the monomial z k is in M p s).

16 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Proof. To prove 1), suppose that f Ms) an write fz)= k N n a kz k. By the Cauchy formula we have a k z k = fu ζ z) ζ k m n ζ ), T n where m n ζ ) is the normalize Lebesgue measure on T n.ifg is a holomorphic polynomial an h L μ n ), an application of Fubini s theorem leas to HUζ f g), h ζ k m n ζ ) = H ak z k g), h..6) T n By Lemmas.11,.1 we see that H ak g) z k L μ n ) T n H Uζ f g) m n ζ )..7) Since H Uζ f g) H f g L μ n ) for all ζ in Tn, it follows that H ak z k is boune an a k z k is in Ms). Therefore, z k Ms) as long as k f z k ). This proves part 1) of the lemma. Suppose now that f M s) an let g q ) be a sequence in A s) which converges weakly to. lim q + H Uζ f g q) L μ n ) =, for all ζ Tn, so that by.7) an the ominate convergence theorem we see that lim q + H ak g z k q ) L μ n ) = an hence z k M s) whenever k f z k ). Therefore part ) of the lemma hols. To establish the remaining part of the lemma, we recall that if T is a compact operator from A s) to L μ n ) then its singular numbers ν q T ), q N,aregivenby ν q T ) := inf A R q T A where R q is the space of all operators from A s) to L μ n ) with finite rank at most q. Assume that f M p s). Then the sequence ν q Hf )) q is in l p. Moreover, there are an orthonormal system u q ) q in A s) an an orthonormal system v q ) q in L μ n ) such that H f = + q= ν q H f ),u q v q,

17 994 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) where the series converges in the operator norm. See Proposition 16.3 in [19]. If q is a positive integer, consier the operator with rank at most q given by q 1 A q := ν j Hf ),u kj v kj j= where for each integer j =,...,q 1 an z C n the functions u kj an v kj are efine by u kj z) := U ζ u j )z) ζ k m n ζ ) an v kj z) := U ζ h j )z) ζ k m n ζ ). T n T n The ominate convergence theorem, combine with.6) an.5), yiels Hak A z k q )g), h = T n + j=q ν j H f ) U ζ g,u j v j U ζ h ζ k m n ζ ). Due to the facts that the sequence ν j Hf )) j is non-increasing an the systems u j ) j an v j ) j are orthonormal it follows that + ν q H f ) U ζ g,u j v j U ζ h ν qhf ) g A s) h L μ n ) j=q for all ζ T n, g A s) an h L μ n ). Hence This implies that H ak A z k q ν q Hf ). ν q H ak ) ν z k q Hf ) showing that a k z k M p s). Consequently, z k M p s) if an only if k f ). The proof of z the lemma is now complete. k Proof of Theorem B. Let f A s). Suppose that Hf is boune an let k Specf ). By Lemma.1 we see that the monomial z k is in Ms). Now Theorem.5 implies that f satisfies 1.1). If Hf is compact resp. in the Schatten class) then a similar argument shows that f satisfies part ) resp. part 3)) of Theorem B. Proof of Corollaries 1.1, 1. an 1.3. As mentione before, the equivalence between 1) an 4) in the statement of each of the corollaries was in establishe in [17] an [15]. The ouble equivalence ) 3) 4) in the statement of each of the corollaries follows from Theorem B. Lemma.13. Suppose that R s =+ an the sequence s satisfies 1.3). Then the function w gw)k s z, w) is in L μ n ) for all holomorphic polynomials g an z C n.

18 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Proof. We first observe that K s z, w) = Therefore, for any α N n an z Cn, C n n 1)! z, w, z C n,w C n. n 1)!! w α K s z, w) ) 1 + ) + n 1)! μ n w) = n 1)!! ) 1 + n 1)! + n 1)!! C n w α z, w μ n w) ) R s z ) 1 + ) + n 1)! = s α + z. n 1)!! t α + μt) Now assumption 1.3) ensures that the latter series converges for all z C n. Lemma.14. Assume that s satisfies 1.). Then the spaces Ms) an M p s), p 1, are Banach spaces an M p s), <p<1, is a quasi-banach space. Proof. We prove the lemma for Ms). Letf q ) q N be a Cauchy sequence in Ms). Without loss of generality we may assume that f q ) = for all n. The sequence Hf q ) q N is a Cauchy sequence of boune operators on A s). Therefore, there is an operator T in A s) such that Hf q ) q N converges to T in the norm operator. Let f := T1) be the conjugate of the image T1) of the constant function 1 uner T. Since Hf q 1) = f q, it follows that showing that f q f L μ n ) = f q T1) L μ n ) = H f q 1) T1) L μ n ) H f q T 1 L μ m ) lim f q f q L μ n ) =..8) Thus f A s). We shall show that the Hankel operator Hf with symbol f is boune. We shall prove that f Ts) an Hf coincies with T on holomorphic polynomials. Let g be a holomorphic polynomial. We first observe by Lemma.13 that for all z C n we have P s f f q )g ) z) f q f L μ n ) gk s z, ) L μ n )

19 996 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) so that by.8) we see that lim q + P s f f q )g)z) =. Since again by.8) we have that lim q + f f q )gz) =, it follows that lim H q + f q Hf )g)z) =. This proves that Tg = Hf g) an hence f Ts) an T = H f. Therefore Ms) is a Banach space. The proof of that M p s) is a Banach space for p 1, an a quasi-banach space for <p<1 is similar. Proof of Theorem C. Suppose that n = 1 an f is as in the hypothesis of Theorem C. A straightforwar calculation appealing to Lemma.1 shows that for all non-negative integers j, k we have Tr H z k H z j ) =, as long as j k an Tr H z ) + k H z k = =k = kr s. s+k + k +k ) k 1 + = +k Writing f = k N a kz k yiels Tr H f H ) f = R s k a k = 1 π k N f z) Az). This proves the first equality of the theorem. Next we prove the secon equality. Writing K s z, w) = k= f k z) f k w), where f k ) is an orthonormal basis of A s), we observe by a stanar argument that for any positive operator T on A s) we have TrT ) = Applying this equality to T = H that Tf k,f k A s) k= = TKs,z),K s,z) A s) Az). f H Tr H f H ) f = f an using the reproucing property of the kernel K s implies fz) fw) Kz,w) Az) an hence completes the proof of the theorem.

20 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) Proof of Theorem D. Follows from Lemmas.1,.1 an Concluing remarks It was prove by the authors [11] that if s := ) is the Stieltjes moment sequence given by s = 1 ) s + n m Γ m where m is positive real number, then the space Ms) is finite-imensional an consists of polynomials of egree at most m. See also [16] an [1] for the one-imensional case. It woul be of interest to characterize Stieltjes moment sequences having this property. Such sequences must not be exponentially boune since, in view of Theorem B, the space Ms) contains all holomorphic polynomials whenever the corresponing sequence s is exponentially boune. Similar iscussion can be invoke relatively to M p s). This issue has been treate in the case of Bergman space on the unit ball. See [7] an [5]. Fix a Stieltjes moment sequence an for a linear operator T on A s), the Berezin transform of T is the function T efine on by where κ z is the normalize reproucing kernel Tz):= Tκ z,κ z, κ z w) := K sw, z). K s z, z) 1 If k N n, we o not know whether the bouneness of the Berezin transform H z k H z k is equivalent to that of the operator H z k. Finally, it woul be of interest to consier further stuy of the general setting of Hankel operators with arbitrary symbols but reasonably efine as waone in the classical Fock space. See [3,4,6,7,3,4] an the references therein. References [1] P. Ahern, E.H. Youssfi, K. Zhu, Compactness of Hankel operators on Hary Sobolev spaces of the polyisk, J. Operator Theory 61 9) [] J. Arazy, S.D. Fisher, J. Peetre, Hankel operators on weighte Bergman spaces, Amer. J. Math ) [3] W. Bauer, Mean oscillation an Hankel operators on the Segal Bargmann space, Integral Equations Operator Theory 5 5) [4] W. Bauer, Hilbert Schmit Hankel operators on the Segal Bargmann space, Proc. Amer. Math. Soc. 13 5) [5] C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups, Gra. Texts in Math., Springer-Verlag, [6] C.A. Berger, L. Coburn, Toeplitz operators on the Segal Bargmann space, Trans. Amer. Math. Soc ) [7] C.A. Berger, L. Coburn, K. Zhu, Toeplitz Operators an Function Theory in n-dimensions, Lecture Notes in Math., vol. 156, Springer, [8] T.M. Bisgaar, Stieltjes moment sequences an positive efinite matrix sequences, Proc. Amer. Math. Soc )

21 998 H. Bommier-Hato, E.H. Youssfi / Journal of Functional Analysis 58 1) [9] T.M. Bisgaar, Z. Sasvári, Stieltjes moment sequences an positive efinite matrix sequences, Math. Nachr ) [1] R.P. Boas, The Stieltjes moment problem for functions of boune variation, Bull. Amer. Math. Soc ) [11] H. Bommier-Hato, E.H. Youssfi, Hankel operators on weighte Fock spaces, Integral Equations Operator Theory 59 7) [1] F. Haslinger, The canonical solution operator to restricte to Bergman spaces an spaces of entire functions, Ann. Fac. Sci. Toulouse Math. 6) 11 ) [13] F. Haslinger, Schröinger operators with magnetic fiels an the canonical solution operator to, J. Math. Kyoto Univ. 46 6) [14] F. Haslinger, B. Helfer, Compactness of the solution operator to in weighte L -spaces, J. Funct. Anal. 43 7) [15] F. Haslinger, B. Lamel, Spectral properties of the canonical solution operator to, J. Funct. Anal. 55 8) [16] W. Knirsch, G. Schneier, Continuity an Schatten von Neumann p-class membership of Hankel operators with antiholomorphic symbols on generalize) Fock spaces, J. Math. Anal. Appl. 3 6) [17] S. Lovera, E.H. Youssfi, Spectral properties of the -canonical solution operator, J. Funct. Anal. 8 4) [18] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas an Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, [19] R. Meise, D. Vogt, Introuction to Functional Analysis, Clarenon Press, Oxfor, [] W. Ruin, Function Theory on the Open Unit Ball in C n, Springer-Verlag, 198. [1] G. Schneier, Hankel operators with antiholomorphic symbols on the Fock space, Proc. Amer. Math. Soc. 13 4) [] T. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse ) 1 1; Ann. Fac. Sci. Toulouse ) [3] K. Stroethoff, Hankel operators in the Fock space, Michigan Math. J ) [4] J. Xia, D. Zheng, Stanar eviation an Schatten class Hankel operators on the Segal Bargmann space, Iniana Univ. Math. J. 53 4) [5] K. Zhu, Hilbert Schmit Hankel operators on the Bergman space, Proc. Amer. Math. Soc ) [6] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 199. [7] K. Zhu, Schatten class Hankel operators on the Bergman space of the unit ball, Amer. J. Math )