On singular values of Hankel operators on Bergman spaces
aa r X i v : . [ m a t h . C A ] F e b ON SINGULAR VALUES OF HANKEL OPERATORS ON BERGMANSPACES
M. BOURASS, O. EL-FALLAH, I. MARRHICH, AND H. NAQOS
Abstract.
In this paper, we study the behavior of the singular values of Hankel operatorson weighted Bergman spaces A ω ϕ , where ω ϕ = e − ϕ and ϕ is a subharmonic function. Weconsider compact Hankel operators H φ , with anti-analytic symbols φ , and give estimatesof the trace of h ( | H φ | ) for any convex function h . This allows us to give asymptotic esti-mates of the singular values ( s n ( H φ )) n in terms of decreasing rearrangement of | φ ′ | / √ ∆ ϕ .For the radial weights, we first prove that the critical decay of ( s n ( H φ )) n is achieved by( s n ( H z )) n . Namely, we establish that if s n ( H φ ) = o ( s n ( H z )), then H φ = 0. Then, weshow that if ∆ ϕ ( z ) ≍ −| z | ) β with β ≥
0, then s n ( H φ ) = O ( s n ( H z )) if and only if φ ′ belongs to the Hardy space H p , where p = β )2+ β . Finally, we compute the asymptoticsof s n ( H φ ) whenever φ ′ ∈ H p . Introduction
Hankel operators are one of the most important classes of bounded linear operators act-ing on spaces of analytic functions. They have many connections with function theory,harmonic analysis, approximation theory, moment problems, spectral theory, orthogonalpolynomials, stationary Gaussian processes, ∂ -operator . . . etc. The book of V. Peller [24]is an acknowledged reference in the classical theory of Hankel operators and their variousapplications. We are interested in the behavior of the singular values of Hankel operatorswith anti-analytic symbols acting on Bergman spaces. The first work in this subject is dueto Axler [3] who described boundedness and compactness of such operators in the classicalBergman space. Right after, Arazy, Fischer and Peetre [2] studied the membership for suchoperators in Schatten classes. They highlighted the existence of a cut-off by proving thatthese operators can not be of finite trace. In [13], Engliˇs and Rochberg described, usingthe Boutet de Monvel-Guillemin theory, all Hankel operators with anti-analytic symbolsthat belongs to the Dixmier class and gave an explicit formula for Dixmier Trace in thiscase. Several authors considered the same problems in other spaces of analytic functions[20, 17, 6, 26, 23, 29, 22]. In this paper, we give the asymptotic behavior of the singular Mathematics Subject Classification.
Primary 47B35, Secondary 30H20, 30C40.
Key words and phrases.
Bergman spaces, Hankel operator, Toeplitz operator, singular values, ¯ ∂ - L minimal solution.Research partially supported by ”Hassan II Academy of Science and Technology”for the first and thesecond authors. values of compact Hankel operators, we describe the class of symbols for which the asso-ciated Hankel operators have the critical decay and compute the asymptotics of singularvalues of Hankel operators associated with this class of symbols.The space of all holomorphic functions on the unit disc D in the complex plane C will bedenoted by Hol( D ). The Lebesgue measure on C is denoted by dA . The standard Bergmanspace A α with α > −
1, consists of holomorphic functions f on D such that k f k α := Z D | f ( z ) | dA α ( z ) < ∞ , where dA α ( z ) := ( α +1) π (1 − | z | ) α dA ( z ).Recall that A α is a reproducing kernel Hilbert space with the kernel K ( z, w ) = 1(1 − wz ) α +2 , z, w ∈ D . The orthogonal projection of L α := L ( dA α ) onto A α will be denoted by P α . Given g ∈ L ( dA α ), the linear transformation H g f = gf − P α ( gf )is a densely defined operator of A α into L α ⊖ A α . The operator H g is called the (big) Hankeloperator with symbol g . For general facts concerning Hankel operators on Bergman spaceswe refer to [2, 29]. In this paper we are interested in Hankel operator H φ with anti-analytic symbol φ . In [2], J. Arazy, S. Fisher and J. Peetre proved that H φ is bounded( resp. compact) on A α if and only if φ belongs to the Bloch space B ( resp. φ belongs tothe little Bloch space B ). This result was first proved by S. Axler [3] in the case α = 0.The membership in Schatten classes of Hankel operators H φ was also studied by J. Arazy,S. Fisher and J. Peetre in [2]. They proved that H φ ∈ S p ( A α ), for p >
1, if and only if φ belongs to the Besov space B p defined by B p = { φ ∈ Hol( D ) , Z D | φ ′ ( z ) | p (1 − | z | ) p − dA ( z ) < ∞} . For p ≤
1, they proved that H φ ∈ S p ( A α ) if and only if H φ = 0 (which means that φ is aconstant). These results were extended by Galanopoulos and Pau in [15] to large Bergmanspaces associated with radial weights.Let ω =: e − ϕ be a weight on D such that ϕ is a regular subharmonic function and let A ω be the weighted Bergman space given by A ω = (cid:8) f ∈ Hol( D ) : k f k ω := (cid:18)Z D | f ( z ) | dA ω ( z ) (cid:19) / < ∞ (cid:9) , dA ω := ωdA. As before, the Hankel operator with anti-analytic symbol φ is the operator H φ : L ( dA ω ) → L ( dA ω ) ⊖ A ω given by H φ f = φf − P ω ( φf ) , INGULAR VALUES OF HANKEL OPERATORS 3 where P ω is the orthogonal projection from L ( dA ω ) onto A ω .For a class of radial rapidily decreasing weights ω ( z ) = e − ϕ ( | z | ) , Galanopoulos and Pauproved in [15] that H φ is bounded (resp. compact) if and only if | φ ′ | ∆ ϕ is bounded (resp.lim | z |→ − | φ ′ ( z ) | ∆ ϕ ( z ) = 0). They also prove, for p >
1, that H φ ∈ S p if and only if | φ ′ | ∆ ϕ ∈ L p/ (∆ ϕdA ).Our goal in this paper is to study the asymptotic behavior of the singular values of H φ .We will consider the class of weights W ∗ which covers all previous examples (see Section2.1). In order to state our main results, we introduce some notations. The reproducingkernel of A ω is denoted by K , τ ω ( z ) = 1 ω / ( z ) k K z k and dλ ω = dAτ ω . It should be noted that in several, but not all, situations τ ω is comparable to 1 / ∆ ϕ . Formore information, see the examples given in Section 2.1.By analogy with the standard case, we write B ω (resp. B ω ) for the space of analyticfunctions φ on D such that sup z ∈ D τ ω ( z ) | φ ′ ( z ) | < ∞ (resp. lim | z |→ − τ ω ( z ) | φ ′ ( z ) | = 0).The following theorem will play an important role in the sequel. Theorem 1.1.
Let ω ∈ W ∗ and let φ ∈ B ω . Let h : [0 , + ∞ [ → [0 , + ∞ [ be an increasingconvex function such that h (0) = 0 . Then there exists B > , which depends only on ω ,such that Z D h (cid:18) B τ ω ( z ) | φ ′ ( z ) | (cid:19) dλ ω ( z ) ≤ Tr ( h ( | H φ | )) ≤ Z D h ( Bτ ω ( z ) | φ ′ ( z ) | ) dλ ω ( z ) . Let us denote by R + φ,ω the decreasing rearrangement of the function τ ω | φ ′ | with respectto dλ ω . Namely, R + φ,ω ( x ) := sup { t ∈ (0 , k τ φ ′ k ∞ ] : R φ,ω ( t ) ≥ x } , where R φ,ω is the distribution function given by R φ,ω ( t ) := λ ω ( { z ∈ D : τ ω ( z ) | φ ′ ( z ) | > t } ) . As a first consequence of Theorem 1.1, we prove that if ρ is an increasing function suchthat ρ ( x ) /x γ is decreasing for some γ <
1, then s n ( H φ ) ≍ /ρ ( n ) ⇐⇒ R + φ,ω ( n ) ≍ /ρ ( n ) . The next result is motivated by a problem raised in [2, 1] . We prove in the followingtheorem that the critical decay of ( s n ( H φ )) n is achieved by the symbol φ = z . Theorem 1.2.
Let ω ∈ W ∗ be such that τ ω is equivalent to a radial function and let φ ∈ B ω . Then M. BOURASS, O. EL-FALLAH, I. MARRHICH, AND H. NAQOS (1) s n ( H φ ) = o ( s n ( H z )) = ⇒ H φ = 0 .(2) s n ( H φ ) = o ( R + z,ω ( n )) = ⇒ H φ = 0 . Note that, in general, it is not difficult to estimate R + z,ω ( n ). For example, if τ ω ( z ) ≍ (1 − | z | ) β with β ≥
0, then R + z,ω ( n ) ≍ n − β β ) . Thus, Theorem 1.2 implies that s n ( H φ ) = o (cid:16) n − β β ) (cid:17) = ⇒ H φ = 0 . Now our goal is to describe the class of functions φ ∈ B ω such that (cid:0) s n ( H φ ) (cid:1) n has thecritical decay. For simplicity, we state our result only in the case τ ω ( z ) ≍ (1 − | z | ) β .As usual, the Hardy space H p , p ≥
1, consists of analytic functions f on D such that k f k pH p := sup ≤ r< π Z π | f ( re iθ ) | p dθ < ∞ . Theorem 1.3.
Let ω ∈ W ∗ be such that τ ω ( z ) ≍ (1 − | z | ) β with β ≥ . Set p = β )2+ β and let φ ∈ B ω . We have s n ( H φ ) = O (1 /n /p ) ⇐⇒ φ ′ ∈ H p . Furthermore, if ω is radial and s n ( H z ) ∼ γ/n p for some γ ∈ (0 , ∞ ) , then s n ( H φ ) ∼ k φ ′ k H p γn p , φ ′ ∈ H p . It should be noted that if τ ω ( z ) ≍ (1 − | z | ) β log − α (2 / − | z | ) with α ≥
0, then R + z,ω ( n ) ≍ log α/ β ( n ) n /p . We prove in Theorem 6.2 that if β >
0, then s n ( H φ ) = O log α/ β ( n ) n /p ! ⇐⇒ φ ′ ∈ H p . However, if β = 0 and α >
0, curiously, the previous result is not valid. This is the subjectof Proposition 6.3.For the standard Bergman spaces A α we have τ ω α ( z ) ≍ (1 − | z | ) . It is known, and canbe easily seen, that s n ( H z ) ∼ √ αn +1 . Hence, Theorem 1.3 says that if φ ∈ B then s n ( H φ ) = O (1 /n ) ⇐⇒ φ ′ ∈ H . And in this case s n ( H φ ) ∼ √ αn +1 k φ ′ k H . This last result improves the results obtained in[8, 13]. Another example is given in Section 8.The paper is organized as follows: In Section 2 we introduce all definitions and notationsthat are used in the rest of the paper. In section 3, we give a description of boundednessand compactness of Hankel operators H φ on A ω . We establish, in Section 4, an upper anda lower estimates of the Trace of h ( | H φ | ). The upper estimate is obtained from H¨ormandertype L ω estimates for ¯ ∂ - equation and from recent estimates obtained by El-Fallah and El INGULAR VALUES OF HANKEL OPERATORS 5
Ibbaoui for Toeplitz operators [10, 9]. The lower estimate is obtained through a sort of lo-cal Berezin transform of H φ . Two direct consequences of trace estimates are obtained by asuitable choice of the convex function h . The first one gives a sharp asymptotic estimates ofthe singular values of compact operators of H φ . The second, presented in Section 5, showsthat the critical decay of the sequence ( s n ( H φ )) n is achieved by the symbol φ = z . InSection 6, we prove the first assertion of Theorem 1.3. The proof is based on Theorem 1.1and on estimates of some maximal non-tangential functions. The second part of Theorem1.3 is given in Section 7. The proof of this result is based on the first part of Theorem 1.3and on a result on asymptotic orthogonality due to A. Pushnitski (see Appendix). Section8 is devoted to an explicit example.Throughout the paper, the notation A . B means that there is a constant c independentof the relevant variables such that A ≤ cB . We write A ≍ B if both A . B and B . A .As usual, the notation u n ∼ v n means that lim n →∞ u n /v n = 1. The letter C will denote anabsolute constant whose value may change at different occurrences. Acknowledgements . The authors are grateful to Evgeni Abakumov for several helpfuldiscussions and suggestions. 2.
Preliminaries
The class of weights W ∗ . Throughout this paper, ω will denote a function from D into ]0 , ∞ [ which is integrable with respect to the Lebesgue measure. The associatedBergman space will be denoted by A ω . We will also assume that ω is bounded below by apositive constant on each compact set of D . This implies that A ω is a reproducing kernelHilbert space. The kernel of A ω will be denoted by K .The orthogonal projection from L ω := L ( D , dA ω ) onto A ω will be designated by P ω . Itcan be represented as follows P ω ( f )( z ) = Z D f ( ζ ) K ( z, ζ ) dA ω ( ζ ) , f ∈ L ω . So, the domain of P ω can be extended to all functions f such that f K z ∈ L ω := L ( D , dA ω ),for all z ∈ D . Let g ∈ L ω such that gK z ∈ L ω , for all z ∈ D . The (big) Hankel operator H g with symbol g is the densely defined operator on A ω defined by H g f = gf − P ω ( gf ) , where f = X ≤ i ≤ n c i K z i , with c i ∈ C and z i ∈ D .The explicit formula for H g is H g f ( z ) = Z D ( g ( z ) − g ( w )) f ( w ) K ( z, w ) dA ω ( w ) , z ∈ D . M. BOURASS, O. EL-FALLAH, I. MARRHICH, AND H. NAQOS
We are interested in this paper in anti-analytic symbols on D , g = φ . In this case a directcomputation gives the following useful formula( H φ K a )( z ) = ( φ ( z ) − φ ( a )) K a ( z ) , z, a ∈ D . (1)First, we recall the definition of the class of weights W introduced in [11]. Let τ ω ( z ) = 1 ω / ( z ) k K z k ω , z ∈ D . Suppose that the reproducing kernel K satisfies the following conditionslim | z |→ − k K z k = ∞ and for every ζ ∈ D , | K ( ζ , z ) | = o ( k K z k ) , | z | → − . (2)We will suppose that τ ω is such that τ ω ( z ) = O (1 − | z | ) , z ∈ D , (3)and that there exists constant η > z, ζ ∈ D satisfying | z − ζ | ≤ ητ ω ( z ), wehave τ ω ( z ) ≍ τ ω ( ζ ) and k K z k ω k K ζ k ω . | K ( ζ , z ) | . (4)If the weight ω satisfies all the previous conditions, we shall say that the weight ω belongsto the class W . Note that W contains all standard weights. For more information, see theexamples listed in [11].The Laplace operator ∆ is given by ∆ = ∂ ¯ ∂ , with ∂ := 12 ( ∂∂x − i ∂∂y ) , ∂ := 12 ( ∂∂x + i ∂∂y ) . Let ω = e − ϕ ∈ W such that ϕ ∈ C ( D ). We shall suppose that τ ω ( z )∆ ϕ ( z ) & , z ∈ D , (5)or, there exist a subharmonic function ψ : D → R + and constants δ > t ∈ ( − , z ∈ D we have τ ω ( z )∆ ψ ( z ) ≥ δ, ∆ ϕ ( z ) ≥ t ∆ ψ ( z ) and | ∂ψ ( z ) | ≤ ∆ ψ ( z ) . (6) Definition 2.1.
We say that ω ∈ W ∗ if ω ∈ W and satisfies (5) or (6). Examples.
In this subsection, we give three examples which will be our referencesthroughout this paper. • Standard Bergman spaces A α : These spaces are associated with ω α ( z ) = ( α +1) π (1 −| z | ) α , with α > −
1. The reproducing kernel of A α is K ( z, w ) = − wz ) α +2 . Then τ ω α ( z ) = r πα + 1 (1 − | z | ) . Clearly, ω α ∈ W . Note that if α >
0, then τ ω α ( z ) ≍ / √ ∆ ϕ , where ϕ = log 1 /ω α .Then ω α satisfies (5) and ω α ∈ W ∗ . It is also clear that if α ∈ ( − , ω α satisfies (6) with t = α and ψ ( z ) = log(1 / − | z | ). Then ω α ∈ W ∗ . INGULAR VALUES OF HANKEL OPERATORS 7 • Harmonically weighted Bergman spaces: In this case we suppose that ω is a positiveharmonic weight. It is proved in [12], that τ ω ( z ) ≍ − | z | and ω ∈ W . One caneasily see that ϕ = log(1 /ω ) is subharmonic and that ω satisfies (6), with t ∈ ( − , ψ ( z ) = log(1 / − | z | ). It should be noted that in general τ ω and 1 / ∆ ϕ are notcomparable. • Large Bergman spaces: The following class of weights was introduced by Hu, Lvand Schuster in [16]. It includes the classes considered in [7, 18, 21]. Let L bethe class of functions τ ∈ lip( D , R ) such that lim | z |→ − τ ( z ) = 0. We denote W theset of weights ω = e − ϕ , where ϕ ∈ C is strictly subharmonic and for which thereexists τ ∈ L such that τ ≍ / √ ∆ ϕ . One can directly see, from [16], that if ω = e − ϕ ∈ W , then ω = e − ϕ ∈ W ∗ and τ ω ≍ / √ ∆ ϕ .2.2. Some inequalities involving convex functions.
The following elementary lemmais proved in [9].
Lemma 2.2.
Let ( a n ) n ≥ , ( b n ) n ≥ be two decreasing sequences such that lim n →∞ a n = lim n →∞ b n =0 and suppose that there exists γ ∈ (0 , such that ( n γ b n ) is increasing. Suppose that thereexists B > such that X n ≥ h ( b n /B ) ≤ X n ≥ h ( a n ) ≤ X n ≥ h ( Bb n ) , for all increasing convex function h . Then a n ≍ b n . We also need the following elementary result. For completness, we include the proof.
Lemma 2.3.
Let
A, p be two real numbers such that < A < p . Let ρ be an increasingfunction such that ρ ( x ) /x A is decreasing and let ( a n ) n be a decreasing sequence such that, X n ≥ h ( a n ) ≤ X n ≥ h (1 /ρ ( n )) , for all increasing function h such that h ( t p ) is convex. Then a n ≤ C ( p, A ) /ρ ( n ) . Proof.
Let δ > h ( t ) = (cid:0) t /p − δ /p (cid:1) + . By hypothesis, X n ≥ (cid:0) a /pn − δ /p (cid:1) + ≤ X n ≥ (cid:0) /ρ /p ( n ) − δ /p (cid:1) + . Then we have c ( p ) X a n ≥ δ a /pn ≤ X ρ ( n ) ≤ /δ /ρ /p ( n ) . Using the fact that n A/p /ρ /p ( n ) is increasing and the fact that A/p <
1, we obtain δ /p Card { n : a n ≥ δ } ≤ C ( p, A ) δ /p Card { n : ρ ( n ) ≤ /δ } . This implies the result. (cid:3)
M. BOURASS, O. EL-FALLAH, I. MARRHICH, AND H. NAQOS
The following elementary lemma will be useful in the proof of Theorem 1.1.
Lemma 2.4.
Let
H, K be two Hilbert spaces and let T : H → K be a compact operator.Suppose there exist ( u n ) n ≥ ⊂ H , ( v n ) n ≥ ⊂ K such that k u n k = k v n k = 1 and X n |h u n , u i| ≤ C k u k and X n |h v n , v i| ≤ C k v k ( u ∈ H, v ∈ K ) , for some C > . Then for any increasing convex function h such that h (0) = 0 , we have X n h ( |h T u n , v n i| ) ≤ C X n h ( s n ( T )) . Proof.
It suffices to use the spectral decomposition of T . (cid:3) Boundedness and compactness of Hankel operators on A ω We need the following L -estimates of solutions of the ∂ -equation due to B. Berndtsson([4], Theorem 3.1). Theorem 3.1 ( B. Berndtsson ) . Let Ω be a domain in C and let ϕ and ψ be subharmonicfunctions in Ω . Assume that ψ satisfies | ∂ψ ( z ) | ≤ ∆ ψ ( z ) , z ∈ Ω . (7) Let s ∈ (0 , . Then, for any function g on Ω , there exists a solution u to the equation ∂u = g such that Z Ω | u | e − ϕ + sψ dA . Z Ω | g | ∆ ψ e − ϕ + sψ dA. Lemma 3.2.
Let ω ∈ W ∗ . There exists C > such that for any function g on D , thereexists a solution u to the equation ∂u = g such that Z D | u ( z ) | ω ( z ) dA ( z ) ≤ C Z D τ ( z ) | g ( z ) | ω ( z ) dA ( z ) . Proof.
Note that if ω satisfies (5), then the result comes directly from ∂ − H¨ormander’stheorem. On the other hand, suppose that ω satisfies condition (6), that is there exist asubharmonic function ψ : D → R + and constant t ∈ ( − ,
0) such that for all z ∈ D wehave 1∆ ψ ( z ) . τ ω ( z ) , t ∆ ψ ( z ) ≤ ∆ ϕ ( z ) and | ∂ψ ( z ) | ≤ ∆ ψ ( z ) . Then by Theorem 3.1, applied to the couple of subharmonic functions ( ϕ − tψ, ψ ) with s = − t , there exists a solution u to the equation ∂u = g such that Z D | u | e − ϕ dA . Z D | g | ∆ ψ e − ϕ dA. Then, we get Z D | u ( z ) | ω ( z ) dA ( z ) . Z D | g ( z ) | τ ω ( z ) ω ( z ) dA ( z ) . (cid:3) INGULAR VALUES OF HANKEL OPERATORS 9
In the sequel k z = K z k K z k will denote the normalized reproducing kernel of A ω . Thefollowing result describes the boundedness of H φ on A ω when ω ∈ W ∗ . Theorem 3.3.
Let ω ∈ W ∗ . Then, the Hankel operator H φ is bounded on A ω if and onlyif φ ∈ B ω . In this case k H φ k ≍ sup z ∈ D τ ω ( z ) | φ ′ ( z ) | , where the implied constants depend onlyon ω .Proof. Suppose that H φ is bounded on A ω . Fix a in D and let δ ≤ η ( η is the constantwhich appears in (4)). By the formula (1), we have Z D | H φ k a ( z ) | ω ( z ) dA ( z ) = Z D | φ ( z ) − φ ( a ) | | k a ( z ) | ω ( z ) dA ( z ) ≥ Z D ( a,δτ ω ( a )) | φ ( z ) − φ ( a ) | | k a ( z ) | ω ( z ) dA ( z ) . Using the fact that | k a ( z ) | ≍ K ( z, z ) and the fact that τ ω ( z ) ≍ τ ω ( a ) when z ∈ D ( a, δτ ω ( a )),we have Z D ( a,δτ ω ( a )) | φ ( z ) − φ ( a ) | k k z k ω ( z ) dA ( z ) ≍ τ ω ( a ) Z D ( a,δτ ω ( a )) | φ ( z ) − φ ( a ) | dA ( z ) . Then we obtain Z D | H φ k a ( z ) | ω ( z ) dA ( z ) & τ ω ( a ) Z D ( a,δτ ω ( a )) | φ ( z ) − φ ( a ) | dA ( z ) . By Cauchy’s representation formula, we get τ ω ( a ) | φ ′ ( a ) | . τ ω ( a ) Z D ( a,δτ ω ( a )) | φ ( z ) − φ ( a ) | dA ( z ) . It follows that τ ω ( a ) | φ ′ ( a ) | . τ ω ( a ) Z D ( a,δτ ω ( a )) | φ ( z ) − φ ( a ) | dA ( z ) . Z D | H φ k a ( z ) | ω ( z ) dA ( z )= k H φ k a k ≤ k H φ k . Hence, φ ∈ B ω .Suppose now that sup z ∈ D τ ω ( z ) | φ ′ ( z ) | < ∞ and let f ∈ A ω . By Lemma 3.2, there existsa solution u to the equation ∂u = φ ′ f (8)satisfying Z D | u ( z ) | ω ( z ) dA ( z ) . Z D τ ω ( z ) | φ ′ ( z ) | | f ( z ) | ω ( z ) dA ( z ) . Therefore, since H φ f is the L ω − minimal solution to the equation (8), we have Z D | H φ f ( z ) | ω ( z ) dA ( z ) . Z D τ ω ( z ) | φ ′ ( z ) | | f ( z ) | ω ( z ) dA ( z ) . (9)Hence k H φ k . sup z ∈ Ω τ ω ( z ) | φ ′ ( z ) | < ∞ . (cid:3) Note that from equation (9), if ( f n ) n is an orthonormal basis of A ω then X n k H φ f n k . X n Z D | f n ( z ) | τ ω ( z ) | φ ′ ( z ) | ω ( z ) dA ( z ) ≍ Z D k K z k τ ω ( z ) | φ ′ ( z ) | ω ( z ) dA ( z )= Z D | φ ′ ( z ) | dA ( z ) . Consequently, if φ ∈ B then H φ ∈ S . We will see in the next paragraph that the converseis also true.We have the following description of the compactness of Hankel operators. Theorem 3.4.
Let ω ∈ W ∗ . Then, the Hankel operator H φ is compact on A ω if and onlyif φ ∈ B ω .Proof. Suppose that H φ is compact. Since ω ∈ W ∗ , k z = K z k K z k converges weakly to 0 when | z | → − . Then lim | z |→ − k H φ k z k = 0. By the proof of Theorem 3.3, we have τ ( z ) | φ ′ ( z ) | . k H φ k z k . This implies that φ ∈ B ω .Conversely, let φ ∈ B ω and let φ r ( z ) = φ ( rz ). Clearly, φ r converges to φ in B ω as r → − .Then by Theorem 3.3 H φ r converges to H φ . Now since φ r ∈ B , H φ r is compact. Hence, H φ is compact. (cid:3) Trace estimates for Hankel operators.
This section is devoted to the proof of Theorem 1.1. Before starting the proof, we recallthe following covering lemma.
Lemma 4.1. ( [11] ) Let X be a subset of C and let τ : X → (0 , ∞ ) be a bounded function.Suppose that there are two constants γ, C > such that, for every z, w ∈ X with | z − w | <γτ ( z ) , we have τ ( z ) C ≤ τ ( w ) ≤ Cτ ( z ) . Set B = C + 1 and let δ ≤ γ/B . There exists asequence ( z n ) n ≥ ⊂ X such that(1) X ⊂ ∪ n ≥ D ( z n , δτ ( z n )) .(2) D ( z n , δτ ( z n ) / C ) ∩ D ( z m , δτ ( z m ) / C ) = ∅ for n = m .(3) For z ∈ D ( z n , δτ ( z n )) we have D ( z, δτ ( z )) ⊂ D ( z n , Bδτ ( z n )) . INGULAR VALUES OF HANKEL OPERATORS 11 (4) ( D ( z n , Bδτ ( z n ))) n is a covering of X of finite multiplicity.Such sequences will be called a ( τ, δ ) − lattice of X . We say that ( R n ) n ∈ L ω if ( R n ) n = ( D ( z n , δτ ω ( z n ))) n and satisfies the conditions (1) − (4)of the above lemma. In the sequel we fix ( R n ) n = ( D ( z n , δτ ω ( z n ))) n ∈ L ω and b > bR n ) n =: ( D ( z n , bδτ ω ( z n ))) n is a covering of D of finite multiplicity.For a positive Borel measure µ on D , the Toeplitz operator associated with µ and definedon A ω is given by T µ f ( z ) = Z D f ( w ) K ( z, w ) ω ( w ) dµ ( w ) , z ∈ D . It is easy to verify that T µ satisfies the following remarkable formula h T µ f, f i = Z D | f ( z ) | ω ( z ) dµ ( z ) . For more properties of Toeplitz operators we refer to [19, 9, 10, 11].The following lemma will be used in the proof of the lower estimate of Tr h ( | H φ | ). Lemma 4.2.
Let ω ∈ W ∗ and let φ ∈ B ω . For any increasing convex function h such that h (0) = 0 , we have X n h ( k χ bR n H φ k z n k ) . X n h ( s n ( H φ )) , where the implied constant depends only on ( bR n ) n and ω .Proof. Let u n = k z n , v n = χ bRn H φ k zn k χ bRn H φ k zn k and remark that k χ bR n H φ k z n k = h H φ u n , v n i . So, itsuffices to prove that the conditions of Lemma 2.4 are satisfied. Indeed, let f ∈ A ω and let dµ = X n k K z n k ω ( z n ) dδ z n . Since ω ∈ W , T µ is bounded [11]. Then we have X n |h u n , f i| = X n | f ( z n ) | k K z n k = h T µ f, f i ≤ k T µ kk f k . Now let g ∈ L ( ωdA ). By Holder inequality, we have X n |h v n , g i| = X n |h χ bR n H φ k z n k χ bR n H φ k z n k , χ bR n g i| ≤ X n k χ bR n g k . k g k . This completes the proof. (cid:3)
Proof of Theorem 1.1.
First, we prove that X n h (cid:0) s n ( H φ ) (cid:1) ≤ Z D h ( C | φ ′ ( z ) | τ ω ( z )) dλ ω ( z ).The equation (9) implies that H ∗ φ H φ . T µ φ . (10)Then, by the monotonicity Weyl’s lemma, we have s n ( H φ ) = λ n ( H ∗ φ H φ ) . λ n ( T µ φ ) . (11) Let ˜ h ( t ) = h ( √ t ). By Theorem 4.5 of [9], we have X h ( s n ( H φ )) = X ˜ h ( s n ( H φ )) ≤ X ˜ h ( Cλ n ( T µ φ )) ≤ X ˜ h ( C µ φ ( R n ) A ( R n ) )= X ˜ h (cid:18) C A ( R n ) Z R n τ ω ( z ) | φ ′ ( z ) | dA ( z ) (cid:19) ≤ X ˜ h (cid:18) C Z R n | φ ′ ( z ) | dA ( z ) (cid:19) . By subharmonicity, we have for all z ∈ R n | φ ′ ( z ) | . A ( R n ) Z bR n | φ ′ ( ζ ) | dA ( ζ ) . Then, Z R n | φ ′ ( z ) | dA ( z ) . A ( R n ) (cid:18)Z bR n | φ ′ ( ζ ) | dA ( ζ ) (cid:19) ≍ (cid:18)Z bR n | φ ′ ( ζ ) | τ ω ( ζ ) dλ ω ( ζ ) (cid:19) . Since h is convex we obtain˜ h (cid:18) C Z R n | φ ′ ( z ) | dA ( z ) (cid:19) ≤ ˜ h C (cid:18)Z bR n | φ ′ ( ζ ) | τ ω ( ζ ) dλ ω ( ζ ) (cid:19) ! ≤ h (cid:18) C Z bR n | φ ′ ( ζ ) | τ ω ( ζ ) dλ ω ( ζ ) (cid:19) ≤ Z bR n h ( C | φ ′ ( ζ ) | τ ω ( ζ )) dλ ω ( ζ ) . Combining these inequalities and the fact that ( bR n ) n is of finite multiplicity, we get X h ( s n ( H φ )) ≤ Z D h ( C | φ ′ ( z ) | τ ω ( z )) dλ ω ( z ) . Now we prove the lower inequality by using Lemma 4.2. We have k χ bR n H φ k z n k ≍ (cid:18)Z bR n | φ ( z ) − φ ( z n ) | k K z k ω ( z ) dA ( z ) (cid:19) / ≍ (cid:18)Z bR n | φ ( z ) − φ ( z n ) | dλ ω ( z ) (cid:19) / . By Cauchy’s formula we have τ ω ( z ) | φ ′ ( z ) | . (cid:18)Z bR n | φ ( ζ ) − φ ( z n ) | dλ ω ( ζ ) (cid:19) / ≍ k χ bR n H φ k z n k , z ∈ R n . INGULAR VALUES OF HANKEL OPERATORS 13
Then Z D h ( τ ω ( z ) | φ ′ ( z ) | dλ ω ( z ) ≍ X n Z R n h ( τ ω ( z ) | φ ′ ( z ) | dλ ω ( z ) ≤ X n h ( C k χ bR n H φ k z n k ) . By Lemma 4.2, we obtain the desired result. (cid:3)
Let us denote by B ω,p = { φ ∈ B ω : Z D ( | φ ′ ( z ) | τ ω ( z )) p dλ ω ( z ) < ∞} ( p > . Note that B ω, = B is the classical Dirichlet space and it doesn’t depend on the weight ω . Note that since τ ω ( z ) = O ((1 − | z | )), B ω,p = { } whenever p ≤
1. Using standardarguments, one can easily prove that an analytic function f on D belongs to B ω,p if andonly if X n (cid:18) µ ( R n ) A ( R n ) (cid:19) p < ∞ , where dµ ( z ) = τ ω ( z ) | f ′ ( z ) | dA ( z ) and ( R n ) n ∈ L ω . In particular, this implies that the family ( B ω,p ) p is increasing.The following result, which extends the main results in [2, 15], is a direct consequence ofTheorem 1.1. Corollary 4.3.
Let p ≥ and let ω ∈ W ∗ . Let φ ∈ B ω . Then H φ ∈ S p ( A ω ) ⇐⇒ φ ∈ B ω,p . As a consequence of Theorem 1.1, we give some estimates of the singular values of com-pact Hankel operators. To this end, let us recall that R φ,ω ( t ) := λ ω ( { z ∈ D : τ ω ( z ) | φ ′ ( z ) | >t } ). Let R + φ,ω be the increasing rearrangement of the function τ ω | φ ′ | . For any increasingfunction h , by a standard computation, we have Z D h ( τ ω ( z ) | φ ′ ( z ) | ) dλ ω ( z ) = Z ∞ R φ,ω ( t ) dh ( t ) . Then, there exists
B > ω such that X n ≥ h (cid:18) B R + φ,ω ( n ) (cid:19) ≤ Z D h ( τ ω ( z ) | φ ′ ( z ) | ) dλ ω ( z ) ≤ X n ≥ h ( B R + φ,ω ( n )) . (12)As a consequence of Theorem 1.1, we obtain the following result. Theorem 4.4.
Let ω ∈ W ∗ and let φ ∈ B ω . Let ρ be an increasing function such ρ ( x ) /x γ is decreasing for some γ ∈ (0 , . Then s n ( H φ ) ≍ /ρ ( n ) ⇐⇒ R + φ,ω ( n ) ≍ /ρ ( n ) . Proof.
By Theorem 1.1 and inequalities (12), there exists
B > h we have X n ≥ h (cid:18) B R + φ,ω ( n ) (cid:19) ≤ X n ≥ h ( s n ( H φ )) ≤ X n ≥ h ( B R + φ,ω ( n )) . By Lemma 2.2, we obtain the desired result. (cid:3)
Theorem 4.5.
Let ω ∈ W ∗ and let φ ∈ B ω . Let ρ be an increasing function such that ρ ( x ) /x A is decreasing for some A > . Suppose that R + φ,ω ( x ) = O (1 /ρ ( x )) , as x → ∞ .Then s n ( H φ ) = O (1 /ρ ( n )) , n → ∞ . Proof.
First, recall that dµ φ ( z ) = τ ω ( z ) | φ ′ ( z ) | dA ( z ) and fix ( R n ) n ∈ L ω . By Inequality(11) it suffices to prove that λ n ( T µ φ ) = O (1 /ρ ( n )). By [9], for any p > B > h such that h ( t p ) is convex and h (0) = 0, wehave X h ( λ n ( T µ φ )) ≤ X h ( Bµ φ ( R n ) /A ( R n )) . Let b > bR n ) n ) ∈ L ω . By subharmonicity of | φ ′ | /p , we have τ ω ( z ) | φ ′ ( z ) | ≤ (cid:18) C ( b, p, ω ) A ( R n ) Z bR n τ /pω ( ζ ) | φ ′ ( ζ ) | /p dA ( ζ ) (cid:19) p , for all z ∈ R n . We obtain from the convexity of h p ( t ) =: h ( t p ) that X h ( Bµ φ ( R n ) /A ( R n )) ≤ X h p (cid:18) C ( b, p, ω ) A ( R n ) Z bR n τ /pω ( ζ ) | φ ′ ( ζ ) | /p dA ( ζ ) (cid:19) ≤ X A ( R n ) Z bR n h ( C ( b, p, ω ) τ ω ( ζ ) | φ ′ ( ζ ) | ) dA ( ζ ) . Z D h ( C ( b, p, ω ) τ ω ( ζ ) | φ ′ ( ζ ) | ) dλ ω ( ζ ) . Combining these inequalities with (12) and the hypothesis that R + φ,ω ( n ) . /ρ ( n ) we obtain X h ( λ n (( T µ φ )) ≤ X n ≥ h ( B (cid:0) R + φ,ω ( n ) (cid:1) ) ≤ X n ≥ h ( B/ρ ( n )) , where B depends on ω, b, p and A . The result comes from Lemma 2.3. (cid:3) The cut-off
In this section we consider weights ω ∈ W ∗ such that τ ω is equivalent to a radial function.We cite as examples, radial weights ω ∈ W ∗ , positive harmonic weights and weights ω = e − ϕ ∈ W such that ∆ ϕ is equivalent to a radial function. Proof of Theorem 1.2.
Suppose that s n ( H φ ) = o ( s n ( H z )). Let δ ∈ (0 ,
1) and let h δ ( t ) =( t − δ ) + . By Theorem 1.1 we have Z D h δ (cid:18) B τ ω ( z ) | φ ′ ( z ) | (cid:19) dλ ω ( z ) ≤ X n ≥ h δ ( s n ( H φ )) . INGULAR VALUES OF HANKEL OPERATORS 15
Let ρ ∈ (1 / ,
1) and put K = Z π | φ ′ ( ρe it ) | dt π . By Jensen’s inequality we have Z D h δ (cid:18) B τ ω ( z ) | φ ′ ( z ) | (cid:19) dλ ω ( z ) ≥ Z h δ (cid:18) B τ ω ( r ) Z π | φ ′ ( re it ) | dt π (cid:19) rdrτ ω ( r ) ≥ Z ρ h δ (cid:18) B τ ω ( r ) Z π | φ ′ ( ρe it ) | dt π (cid:19) drτ ω ( r )= Z ρ (cid:18) KB τ ω ( r ) − δ (cid:19) + drτ ω ( r ) . Suppose that φ ′ = 0, then K >
0. For τ ω ( r ) ≥ BK δ , we have (cid:0) KB τ ω ( r ) − δ (cid:1) + ≥ K B τ ω ( r ).Then we obtain Z D h δ (cid:18) B τ ω ( z ) | φ ′ ( z ) | (cid:19) dλ ω ( z ) ≥ K B Z { r ∈ ( ρ, τ ω ( r ) ≥ BK δ } drτ ω ( r ) . (13)Let ε ∈ (0 , K/ B ) and let N be such that for n ≥ N we have s n ( H φ ) ≤ εs n ( H z ). UsingTheorem 1.1, we have X n ≥ h δ ( s n ( H φ )) ≤ X n
Let ω ∈ W ∗ be such that τ ω ( z ) ≍ (1 − | z | ) β ν (log( −| z | )) , where β ≥ and ν is a monotone function such that ν (2 t ) ≍ ν ( t ) . Let φ ∈ B ω and let p = β )2+ β . If s n ( H φ ) = o n p ν β (log n ) ! , then φ ′ = 0 . Proof.
It is not difficult to verify that Z δ dtt β ν (log 1 /t ) ≍ δ β ν (log 1 /δ ) . Then, since τ ω ( z ) ≍ (1 − | z | ) β ν (log( −| z | ), we have R z,ω ( t ) = Z { z ∈ D : τ ω ( z ) ≥ t } dA ( z )(1 − | z | ) β ν (log 1 / (1 − | z | )) ≍ t β )2+ β ν β (log 1 /t ) . This implies that R + z,ω ( n ) ≍ n p ν β (log n ) . By the second assertion of Theorem 1.2, we get φ ′ = 0. (cid:3) Note that if ω is radial then H ∗ z H z (cid:18) z n k z n k (cid:19) = (cid:18) k z n +1 k k z n k − k z n k k z n − k (cid:19) z n k z n k =: m ω ( n ) z n k z n k , n ≥ . So, the sequence of the singular values of H z is the decreasing rearrangement of the sequence( m ω ( n )) n ≥ .For the standard Bergman spaces A α , it is easy to see that k z n k = Γ( n + 1)Γ( α + 2)Γ( n + α + 2) and m ω α ( n ) = α + 1( n + α + 1)( n + α + 2) . where Γ( x ) = Z ∞ t x − e − t dt denotes the Gamma function.Then s n ( H z ) ∼ √ α + 1 n + 1 . For larger Bergman spaces we have the following result.
Proposition 5.2.
Let ω ∈ W ∗ and let β > . Suppose that τ ω ( z ) ≍ (1 −| z | ) β ν (log( −| z | )) ,where ν is a monotone function which satisfies ν (2 t ) ≍ ν ( t ) . Then s n ( H ¯ z ) ≍ n p ν β (log n ) , where p = 2(1 + β )2 + β . Proof.
From the hypothesis, we have R + z,ω ( n ) ≍ n p ν β (log n ) . By Theorem 4.4, we obtainthe result. (cid:3) INGULAR VALUES OF HANKEL OPERATORS 17 Critical decay
In this section we describe the class of functions φ ∈ B ω such that s n ( H φ ) = O ( s n ( H z )).This kind of problem was first examined by Arazy, Fisher and Peetre for standard Bergmanspaces [2]. They proved that if φ belongs to the Besov space B given by B := { f ∈ Hol( D ) , Z D | f ′′ ( z ) | dA ( z ) < ∞} , then σ n ( H φ ) = O (log( n + 1)) , where σ n ( H φ ) := n X j =1 s j ( H φ ) . They also proved, in the same paper, that the converse is false. In [13], M. Engliˇs and R.Rochberg gave a complete answer to this problem for the classical Bergman space. Theyproved, by using Boutet de Monvel-Guillemin theory, that s n ( H φ ) = O (1 /n ) if and only if φ ′ ∈ H . This result was extended by R. Tytgat [27, 28] to standard Bergman spaces A α .In what follows we study this problem in more general situations. Our approach is basedon Theorem 1.1. Proposition 6.1.
Let ω ∈ W ∗ be a weight such that τ ω ( z ) ≍ (1 − | z | ) β ν (log( −| z | ) where β ≥ and ν is a monotone function such that ν (2 t ) ≍ ν ( t ) . Let φ ∈ B ω and let p = β )2+ β . Then s n ( H φ ) = O n p ν β (log n ) ! = ⇒ φ ′ ∈ H p . Proof.
Suppose that s n ( H φ ) ≤ Cn p ν β (log n ) . Since τ ω ( z ) ≍ (1 − | z | ) β ν (log( −| z | ), wehave R z,ω ( t ) ≍ t p ν β (log(1 /t )) and R + z,ω ( n ) ≍ n p ν β (log( n )) . Then for every increasing function h , we have ∞ X n =1 h B n p ν β (log( n )) ! ≤ Z D h ( τ ω ( z )) dλ ω ( z ) ≤ ∞ X n =1 h B n p ν β (log( n )) ! . where B doesn’t depend on h . If in addition h is convex and h (0) = 0, then by Theorem1.1we obtain Z D h ( τ ω ( z ) | φ ′ ( z ) | ) dλ ω ( z ) . ∞ X n =1 h (cid:0) Bs n ( H φ ) (cid:1) . ∞ X n =1 h (cid:0) BC R + z,ω ( n ) (cid:1) . Z D h ( B BCτ ω ( z )) dλ ω ( z ) . Let ε ∈ (0 ,
1) and put p ε = (1 − ε ) p + ε . Note that if β = 0 (i.e. p = 1) then p ε = 1 and if β > < p ε < p . Note that in the two cases we have Z drτ − p ε ω ( r ) = ∞ . The last inequality, with h ( t ) = h δ ( t p ε ), becomes Z D h δ ( τ p ε ω ( z ) | φ ′ ( z ) | p ε ) dλ ω ( z ) . Z D h δ ( K p ε τ p ε ω ( z )) dλ ω ( z ) , where K = CBB . Using the convexity of h δ , we get Z h δ (cid:18) τ p ε ω ( r ) Z π | φ ′ ( re it ) | p ε dt π (cid:19) drτ ω ( r ) ≤ Z D h δ ( τ p ε ω ( z ) | φ ′ ( z ) | p ε ) dλ ω ( z ) . Z D h δ ( K p ε τ p ε ω ( z )) dλ ω ( z ) . Suppose that there exists ρ ∈ (0 ,
1) such that k φ ′ ρ k p ε ≥ K . We have Z ρ h δ (cid:0) τ p ε ω ( r ) k φ ′ ρ k p ε p ε (cid:1) drτ ω ( r ) . Z D h δ ( K p ε τ p ε ω ( z )) dλ ω ( z ) . Now, using the fact that h δ ( t ) ≥ t/ t ≥ δ , we get k φ ′ ρ k p ε p ε Z { r ∈ ( ρ, τ pεω ( r ) k φ ′ ρ k pεpε ≥ δ } drτ − p ε ω ( r ) . K p ε Z { r ∈ (0 , τ pεω ( r ) K pε ≥ δ } drτ − p ε ω ( r ) . Since k φ ′ ρ k p ε ≥ K , { r ∈ ( ρ,
1) : τ p ε ω ( r ) K p ε ≥ δ } ⊂ { r ∈ ( ρ,
1) : τ p ε ω ( r ) k φ ′ ρ k p ε p ε ≥ δ } . Then k φ ′ ρ k p ε p ε Z { r ∈ ( ρ, τ pεω ( r ) K pε ≥ δ } drτ − p ε ω ( r ) . K p ε Z { r ∈ (0 , τ pεω ( r ) K pε ≥ δ } drτ − p ε ω ( r ) . Recall that Z drτ − p ε ω ( r ) = ∞ . We obtain, when δ goes to 0 + , that k φ ′ ρ k p ε . K . When ε goes to 0, we get k φ ′ ρ k p . K . This proves that φ ′ ∈ H p . (cid:3) Proof of the first assertion of Theorem 1.3.
Let p = p β (:= β )2+ β ). Suppose that s n ( H φ ) = O (1 /n /p ). By Proposition 6.1, with ν = 1, we have φ ′ ∈ H p .For the converse, by Theorem 4.5, it suffices to prove that R + φ,ω ( x ) = O (1 /x /p ), whenever φ ′ ∈ H p . To this end, let U be the non-tangential maximal function of φ ′ . Since φ ′ ∈ H p INGULAR VALUES OF HANKEL OPERATORS 19 we have U ∈ L p and k U k p . k φ ′ k p (even if p = 1). We have R φ,ω ( t ) = λ ω ( { z ∈ D : τ ω ( z ) | φ ′ ( z ) | ≥ t } ) ≤ λ ω (cid:0)(cid:8) re iθ ∈ D : τ ω ( r ) U ( e iθ ) ≥ t (cid:9)(cid:1) ≍ Z { re iθ ∈ D : τ ω ( r ) U ( e iθ ) ≥ t } drdθπ (1 − r ) β . t p Z π U p ( e iθ ) dθ π . t p k φ ′ k pp . This is equivalent to R + φ,ω ( x ) . k φ ′ k p x /p . The proof is complete. (cid:3) Now, we study the converse of Proposition 6.1, when τ ω ( z ) ≍ (1 − | z | ) β ν (log( −| z | )).In the following result we consider the case β >
0. The case β = 0, will be discussed rightafter. Theorem 6.2.
Let ω ∈ W ∗ be a weight such that τ ω ( z ) ≍ (1 − | z | ) β ν (log( −| z | )) ,where β > and ν is a monotone function such that ν (2 t ) ≍ ν ( t ) . Let φ ∈ B ω and let p = β )2+ β . Then s n ( H φ ) = O n p ν β (log n ) ! ⇐⇒ φ ′ ∈ H p . In this case we have s n ( H φ ) . k φ ′ k p n p ν β (log n ) , where the involved constant doesn’t depend on φ .Proof. By Proposition 6.1, It remains to prove that if φ ′ ∈ H p then s n ( H φ ) = O (cid:18) n p ν β (log n ) (cid:19) .We will proceed as in the proof of Theorem 1.3. Without loss of generality, suppose that k φ ′ k p = 1. Our goal is to show that R φ,ω ( t ) = O t p ν β (log(1 /t )) ! , where the implied constant doesn’t depend on φ .We have | φ ′ ( rζ ) | ≤ − r ) p = 1(1 − r ) β β ) , r ∈ (0 ,
1) and ζ ∈ T . (15) This implies that | φ ′ ( rζ ) | τ ω ( r ) . (1 − r ) (2+ β ) β β ) ν (log(1 / − r )) . Then there exists r ∈ (0 , ω , such that | φ ′ ( rζ ) | τ ω ( r ) ≤ (1 − r ) (2+ β ) β β ) , r ∈ ( r , . So, if | φ ′ ( rζ ) | τ ω ( r ) ≥ t then r ≤ r t where r t is given by (1 − r t ) (2+ β ) β β ) = t .Let U t be the non-tangential maximal function associated with z ∈ D → φ ′ ( r t z ). Byinequality (15), we have | φ ′ ( rζ ) | ≤ U t ( ζ ) ≤ − r t ) /p = 1 t /β , r ≤ r t . (16)For t small enough, we have R φ,ω ( t ) = λ ω ( { re iθ ∈ D : τ ω ( r ) | φ ′ ( re iθ ) | ≥ t } ) . Z { re iθ : r ∈ (0 ,r t ) , τ ω ( r ) U t ( e iθ ) ≥ t } drτ ω ( r ) dθ . Z π Z { r ∈ (0 ,r t ): τ ω ( r ) ≥ tUt ( eiθ ) } drτ ω ( r ) dθ . Z π R z,ω (cid:18) tU t ( e iθ ) (cid:19) dθ. Since τ ω ≍ (1 − r ) β ν (log(1 / − r )), we have R z,ω ( t ) ≍ t p ν β (log 1 /t ) . First, note thatif U t ( e iθ ) ≤ t / then R z,ω (cid:16) tU t ( e iθ ) (cid:17) ≤ R z,ω ( t / ) ≤ R z,ω ( t ). Otherwise, from equation (16),we get R z,ω (cid:18) tU t ( e iθ ) (cid:19) ≍ U pt ( e iθ ) t p ν β (log U t ( e iθ ) /t ) . U pt ( e iθ ) t p ν β (log 1 /t ) . Combining these inequalities we obtain R φ,ω ( t ) . k U t k pp t p ν β (log 1 /t ) . t p ν β (log 1 /t ) , where the involved constant doesn’t depend on φ ′ . By Theorem 4.5, we obtain the desiredresult. (cid:3) Now, we will prove that the previous result is not true when β = 0 which is somewhatunexpected. Proposition 6.3.
Let ω ∈ W ∗ such that τ ω ( z ) ≍ (1 −| z | )log α (1 / −| z | ) with α > . Then for ν ∈ ] α +12 α +1 , , there exists φ ∈ B ω such that φ ′ ∈ H and s n ( H φ ) ≍ n ν . INGULAR VALUES OF HANKEL OPERATORS 21
Proof.
Let γ ∈ (1 , α + 1) be such that ν = α + γ α +1 . Let φ be such that φ ′ ( z ) = 1(1 − z ) log γ ( e − z ) , z ∈ D . Since γ > , φ ′ ∈ H . Write R φ,ω = R + R , where R ( t ) = λ ω ( { re iθ ∈ D : 1 − r > | θ | and τ ω ( r ) | φ ′ ( re iθ ) | ≥ t } )and R ( t ) = λ ω ( { re iθ ∈ D : 1 − r ≤ | θ | and τ ω ( r ) | φ ′ ( re iθ ) | ≥ t } ) . Clearly, we have R ( t ) ≍ λ ω ( { re iθ ∈ D : 1 − r > | θ | and 1log α + γ ( e/ (1 − r )) ≥ t } ) ≍ t α +1 α + γ . Similarly, R ( t ) ≍ λ ω ( { re iθ ∈ D : 1 − r ≤ | θ | and 1 − r log α (1 / − r ) 1 | θ | log γ ( e/ | θ | ) ≥ t } ) ≍ t α +1 α + γ . Then, R φ,ω ( t ) ≍ t α +1 α + γ . By Theorem 4.4, we obtain that s n ( H φ ) ≍ n ν . The proof iscomplete. (cid:3) Asymptotics
Now, we will precise the results obtained in Section 6 when s n ( H z ) is regular. The mainresult of this section is the following theorem. Theorem 7.1.
Let ω ∈ W ∗ be a radial weight such that τ ω ( z ) ≍ (1 −| z | ) β ν (log( −| z | )) ,where β > and ν is a monotone function such that ν (2 t ) ≍ ν ( t ) . Suppose that s n ( H z ) ∼ γn p ν β (log n ) , where γ > and p = β )2+ β . Then, for φ ′ ∈ H p , we have s n ( H φ ) ∼ γn p ν β (log n ) k φ ′ k p . To prove this result, we will introduce the following functionals (see [5, 25]). Let T be acompact operator between two Hilbert spaces. Let n ( s, T ) be the singular values countingfunction given by n ( s, T ) = { n : s n ( T ) ≥ s } , s > . The class of strictly increasing continuous functions ψ : (0 , + ∞ ) → (0 , + ∞ ) such that ψ (0) = 0 and such that ψ ( αt ) ∼ α p ψ ( t ) , ( t → + ) , for some p >
0, will be denoted by C p . Let D ψ ( T ) := lim sup s → + ψ ( s ) n ( s, T ) and d ψ ( T ) := lim inf s → + ψ ( s ) n ( s, T ) . From these definitions it is easy to see that if ψ ∈ C p , then d ψ ( T ) = D ψ ( T ) ∈ (0 , ∞ ) = ⇒ s n ( T ) ∼ D ψ ( T ) /p ψ − (1 /n ) . For more definitions and properties of these functionals see Section 8.First, we need some results on the operator H z P ω M g , where M g denotes the multiplica-tion operator defined on L ( dA ω ).For an arc δ = { e iθ : θ ≤ θ < θ } ⊂ T , R δN will denote R δN = { z = re iθ : 0 < − r ≤ π/N, θ ≤ θ < θ } , N = 1 , , ... The proof of the following results is similar to that one given by A. Pushnitski in [25].
Lemma 7.2.
Let ω ∈ W ∗ be a radial weight. Let ψ ∈ C p such that D ψ ( H z ) is finite. Thefollowing are true(1) Let g be a bounded function on D . We have D ψ ( H z P ω M g ) ≤ k g k p ∞ D ψ ( H z ) . (2) Let δ , δ be two arcs of T such that δ ∩ δ = ∅ . Then (cid:18) H z P ω M χ Rδ N (cid:19) ∗ (cid:18) H z P ω M χ Rδ N (cid:19) ∈ ∩ p> S p . (3) Let δ ⊂ ∂ D be an arc such that | δ | < π . Then D ψ ( H z P ω χ R δN ) ≤ | δ | D ψ ( H z ) . (4) Let δ , δ be two arcs of T such that δ ∩ δ is reduced to one point. Then (cid:18) H z P ω M χ Rδ N (cid:19) ∗ (cid:18) H z P ω M χ Rδ N (cid:19) ∈ Σ ψ ◦√ . Proof.
Since g is bounded, M g is bounded on L ω and k M g k = k g k ∞ . Then n ( s, H z P ω M g ) ≤ n ( s, k M g k H z ) = n ( s, k g k ∞ H z ), for all s >
0. Hence, by Proposition 8.1, we get D ψ ( H z P ω M g ) ≤ D ψ ( k g k ∞ H z ) = k g k p ∞ D ψ ( H z ) . This proves the first assertion.The same proof as that proposed by Pushnitski in [25], gives the second assertion.To prove the third assertion, let N ∈ N ∗ such that πN +1 ≤ | δ | < πN and let δ n := e πin/N δ, n = 1 , ..., N. INGULAR VALUES OF HANKEL OPERATORS 23
Let R nN := R δ n N . Since g := N X n =1 χ R nN ≤
1, we have D ψ ( H z ) ≥ D ψ ( H z P ω M g ) = D ψ N X n =1 H z P ω M χ RnN ! . For i = j we have (cid:16) H z P ω M χ RiN (cid:17) (cid:18) H z P ω M χ RjN (cid:19) ∗ = 0. Since the closures of R iN and R jN are disjoint, we have (cid:16) H z P ω M χ RiN (cid:17) ∗ (cid:18) H z P ω M χ RjN (cid:19) ∈ Σ ψ ◦√ . . Further, since ω is radial, the operators H z P ω M χ RjN are unitarily equivalent. It followsfrom Theorem 8.3 that D ψ ( H z ) ≥ D ψ N X n =1 H z P ω M χ RnN ! = N D ψ ( H z P ω M χ RδN ) . Therefore, D ψ ( H z P ω M χ RδN ) ≤ /N D ψ ( H z ) ≤ | δ | D ψ ( H z ).The last assertion is a consequence of the second and the third assertion. Indeed, let ε > δ = { e iθ : θ ≤ θ < θ } and δ = { e iθ : θ ≤ θ < θ } and let δ ( ε ) = { e iθ : θ ≤ θ < θ − ε } . By (2), (cid:18) H z P ω M χ Rδ ε ) N (cid:19) ∗ (cid:18) H z P ω M χ Rδ N (cid:19) ∈ ∩ p> S p . ByCorollary 8.2, we get D ψ ◦√ . (cid:18)(cid:18) H z P ω M χ Rδ N (cid:19) ∗ (cid:18) H z P ω M χ Rδ N (cid:19)(cid:19) = D ψ ◦√ . (cid:18)(cid:18) H z P ω M χ Rδ N \ Rδ ε ) N (cid:19) ∗ (cid:18) H z P ω M χ Rδ N (cid:19)(cid:19) . Applying Proposition 8.1, we have D ψ ◦√ . (cid:18)(cid:18) H z P ω M χ Rδ N \ Rδ ε ) N (cid:19) ∗ (cid:18) H z P ω M χ Rδ N (cid:19)(cid:19) ≤ D ψ ( H z P ω M χ Rδ N \ Rδ ε ) N ) D ψ ( H z P ω M χ Rδ N ) . εD ψ ( H z ) . Letting ε to 0, we obtain D ψ ◦√ . (cid:18)(cid:18) H z P ω M χ Rδ N (cid:19) ∗ (cid:18) H z P ω M χ Rδ N (cid:19)(cid:19) = 0, as required. (cid:3) Proposition 7.3.
Let ω ∈ W ∗ be a radial weight. Let ψ ∈ C p such that < d ψ ( H z ) ≤ D ψ ( H z ) < ∞ . • Let δ ⊂ ∂ D be an arc such that | δ | = πN , where N ∈ N ∗ . Then D ψ ( H z P ω M χ RδN ) = 1
N D ψ ( H z ) and d ψ ( H z P ω M χ RδN ) = 1
N d ψ ( H z ) . • Let g be a continuous function on D . We have k g k pp d ψ ( H z ) ≤ d ψ ( H z P ω M g ) ≤ D ψ ( H z P ω M g ) ≤ k g k pp D ψ ( H z ) . Proof.
Let R N = { z = re iθ ∈ D : r > − /N } ,R kN = { z = re iθ ∈ R N : 2 πkN ≤ θ < π ( k + 1) N } , k = 0 , ..., N − . And let h := N − X k =0 χ R kN = 1 − χ D \ R N . Hence, by Corollary 8.2, Lemma 7.2 and Theorem 8.3, we have D ψ ( H z ) = D ψ ( H z P ω M h )= D ψ N − X k =0 H z P ω M χ RkN ! = N − X k =0 D ψ ( H z P ω M χ RkN )= N D ψ ( H z P ω M χ RδN ) . Then D ψ ( H z P ω M χ RδN ) = N D ψ ( H z ). Similarly, we obtain d ψ ( H z P ω M χ RδN ) = N d ψ ( H z ).Let ξ k be the center of the arc R kN ∩ ∂ D and let g N = N − X k =0 g ( ξ k ) χ R kN . By Proposition 8.1, we have D p +1 ψ ( H z P ω M g ) ≤ D p +1 ψ ( H z P ω M g N ) + D p +1 ψ ( H z P ω M g − g N )On one hand, by Lemma 7.2, we have D ψ ( H z P ω M g − g N ) ≤ k g − g N k p ∞ D ψ ( H z ) . On the other hand, by Lemma 7.2, Theorem 8.3 and the first assertion of this proposition,we obtain D ψ ( H z P ω M g N ) ≤ N − X k =0 D ψ ( g ( ξ k ) H z P ω M χ RkN )= N − X k =0 | g ( ξ k ) | p D ψ ( H z P ω M χ RkN )= N N − X k =0 | g ( ξ k ) | p D ψ ( H z ) . Combining these inequalities and letting N going to ∞ , we obtain D ψ ( H z P ω M g ) ≤ k g k pp D ψ ( H z ) . The lower estimates can be obtained by the same arguments. (cid:3)
INGULAR VALUES OF HANKEL OPERATORS 25
Let Φ : R D × R D → C be an analytic function, where R >
1. Let A Φ : A ω → L ω be theoperator defined by A Φ f ( z ) = Z D ( z − ξ ) K ( z, ξ )Φ( z, ξ ) f ( ξ ) ω ( ξ ) dA ( ξ ) , z ∈ D . (17)And let A be the operator given by Af ( z ) = Z D ( z − ξ ) K ( z, ξ ) f ( ξ ) ω ( ξ ) dA ( ξ ) , z ∈ D . (18)Now, we introduce notations which will be used in the proof of the next lemma. Let r ∈ (0 ,
1) and let J r be the embedding operator from A ω to L ( ωχ r D dA ). Note that T r := J ∗ r J r is the Toeplitz operator with symbol χ r D . Namely, T r f = Z r D f ( ζ ) K ( ., ζ ) ω ( ζ ) dA ( ζ ) . By [9], there exists C = C ( ω ) such that X n ≥ λ pn ( T r ) ≤ Cp Z r dρτ ω ( ρ ) , p ∈ (0 , . This implies that λ n ( T r ) ≤ inf p ∈ (0 , (cid:18) Cpn Z r dρτ ω ( ρ ) (cid:19) p = exp − e − C Z r dρτ ω ( ρ ) n . If we suppose that τ ω ( z ) ≍ (1 − | z | ) β ν (log(1 / − | z | )), then λ n ( T r ) ≤ exp (cid:0) − C ( ω )(1 − r ) β ν (log(1 / − r )) n (cid:1) . In particular we have λ n ( T r ) = O (cid:0) exp (cid:0) − C ( ω, ε )(1 − r ) β − ε n (cid:1)(cid:1) , ∀ ε > . (19) Lemma 7.4.
Let ω ∈ W ∗ be a radial weight such that τ ω ( z ) ≍ (1 − | z | ) β ν (log( −| z | )) ,where β ≥ and ν is a monotone function such that ν (2 t ) ≍ ν ( t ) . Then, for all ε > , wehave s n ( A Φ ) = O (cid:18) n /p − ε (cid:19) , p = 2(1 + β )2 + β . Proof.
First, we prove the result for A which corresponds to Φ = 1. Remark that Af = H ¯ z f − H ¯ z P ω ¯ zf ∈ A ⊥ ω . Then Af is the L ω -minimal solution of ¯ ∂u = 2 H ¯ z f . Applying Lemma 3.2 twice, we get k Af k . Z D | H ¯ z f ( z ) | τ ω ( z ) ω ( z ) dA ( z ) . Z r D | H ¯ z f ( z ) | ω ( z ) dA ( z ) + τ ω ( r ) Z D \ r D | H ¯ z f ( z ) | ω ( z ) dA ( z ) . Z r D | f ( z ) | ω ( z ) dA ( z ) + Z r D | P ω ¯ zf ( z ) | ω ( z ) dA ( z ) + τ ω ( r ) Z D | f ( z ) | τ ω ( z ) ω ( z ) dA ( z ) . k J r f k + k J r P ω ¯ zf k + τ ω ( r ) h T τ ω f, f i . Then we obtain A ∗ A . T r + ( P ω ¯ z ) ∗ T r P ω ¯ z + τ ω ( r ) T τ ω This implies that s n ( A ) . λ n ( T r ) + τ ω ( r ) λ n ( T τ ω ) . (20)Since Z drτ − p − εω ( r ) < ∞ , by [9], T τ ω ∈ S p + ε for every ε >
0. Then, λ n ( T τ ω ) = O (cid:18) n p − ε (cid:19) , for all ε > . (21)Combining inequalities (19), (20) and (21), we obtain s n ( A ) . inf r ∈ (0 , (cid:18) exp (cid:0) − C ( ω, ε )(1 − r ) β − ε n (cid:1) + τ ω ( r ) n p − ε (cid:19) , for all ε > . For a suitable choice of r , we obtain the result for A .To complete the proof of the Lemma we use the argument given by M. Dostanic in [8].Suppose that Φ is analytic and bounded by M on R D × R D . Clearly, we haveΦ( z, ζ ) = X k ≥ Φ ( k ) ( z, k ! ζ k and | Φ ( k ) ( z, | k ! ≤ MR k . And since A Φ f = X k ≥ Φ ( k ) ( z, k ! AP ω ¯ ζ k f , we obtain s ( N +2) m ( A Φ ) ≤ N X k ≥ | Φ ( k ) ( z, | k ! s m ( A ) + X k>N | Φ ( k ) ( z, | k ! k A k ≤ MR − s m ( A ) + k A k ) . Now for
N m = n and m ∼ n − ε , we get the result. (cid:3) Proof of Theorem 7.1.
Let ψ ( t ) = γ p t p ν p/ β (log 1 /t ). Since s n ( H z ) ∼ γn p ν β (log n ) , D ψ ( H z ) = d ψ ( H z ) = 1. Note that s n ( H ¯ φ ) ∼ s n ( H z ) k φ ′ k p ⇐⇒ D ψ ( H ¯ φ ) = d ψ ( H ¯ φ ) = k φ ′ k pp . INGULAR VALUES OF HANKEL OPERATORS 27
First, suppose that φ is analytic in a neighborhood of D . Then there exist R > R D × R D such that φ ( z ) − φ ( w ) = ( z − w ) φ ′ ( w ) + ( z − w ) Φ( z, w ) , z, w ∈ R D . We have H φ P ω = H z P ω ¯ φ ′ + A Φ P ω . By Lemma 7.4 s n ( A Φ P ω ) = o ( s n ( H z P ω )) , n → ∞ . So, D ψ ( A Φ P ω ) = 0 . By Corollary 8.2, we deduce that D ψ ( H φ ) = D ψ ( H z P ω ¯ φ ′ ) and d ψ ( H φ ) = d ψ ( H z P ω ¯ φ ′ ) . We obtain, by Proposition 7.3, that d ψ ( H z ) k φ ′ k pp ≤ d ψ ( H φ ) ≤ D ψ ( H φ ) ≤ D ψ ( H z ) k φ ′ k pp . Since D ψ ( H z ) = d ψ ( H z ) = 1, we obtain d ψ ( H φ ) = D ψ ( H φ ) = k φ ′ k pp . Now, suppose that φ ′ ∈ H p . By Theorem 6.2, we have s n ( H ¯ φ − ¯ φ r ) ≤ C k φ ′ − φ ′ r k p s n ( H z ) , n ≥ . This implies that D ψ ( H ¯ φ − ¯ φ r ) ≤ C p k φ ′ − φ ′ r k pp D ψ ( H z ). Then we have | D ψ ( H φ ) p +1 − (cid:0) D ψ ( H z ) k φ ′ r k pp (cid:1) p +1 | = | D ψ ( H φ ) p +1 − D ψ ( H φ r ) p +1 |≤ D ψ ( H φ − φ r ) p +1 ≤ (cid:0) C p k φ ′ − φ ′ r k pp D ψ ( H z ) (cid:1) p +1 . When r → − , we obtain D ψ ( H φ ) = D ψ ( H z ) k φ ′ k pp = k φ ′ k pp . With the same arguments wehave d ψ ( H φ ) = d ψ ( H z ) k φ ′ k pp . The proof is complete. (cid:3) Similarly, one can prove the following result which corresponds to the situation β = 0and ν ≍ Theorem 7.5.
Let ω ∈ W ∗ be a radial weight such that τ ω ( z ) ≍ (1 − | z | ) . Suppose that s n ( H z ) ∼ γn , where γ > . Then, for φ ′ ∈ H , we have s n ( H φ ) ∼ γn k φ ′ k . Proof of the second assertion of Theorem 1.3.
It suffices to combine Theorem 7.5 and The-orem 7.1 with ν = 1. (cid:3) An example
For a radial weight ω , the sequence of the singular values of the Hankel operator H z on A ω is the decreasing rearrangement of the sequence (cid:18) || z n +1 || || z n || − || z n || || z n − || (cid:19) / ! n . Recall that for the standard case ω α ( z ) = ( α + 1)(1 − | z | ) α , we have s n ( H z ) ∼ √ α +1 n . Inthis section, we consider the weight ω ( z ) = exp − α (log | z | ) β ! , α, β > . We have || z n || = Z D | z | n exp − α (log | z | ) β ! dA ( z )= Z r n exp (cid:18) − α (log r ) β (cid:19) rdr = Z + ∞ exp (cid:16) − ( n + 1) x − αx β (cid:17) dx. Let x n := (cid:0) αβn +1 (cid:1) / β be the minimum of the function ( n + 1) x + αx β . After the change ofvariable u = x − x n x n , we get || z n || = x n exp (cid:18) − ( n + 1) x n − αx βn (cid:19) Z + ∞− exp (cid:18) − αx βn h ( u ) (cid:19) du, where h ( u ) = βu + u ) β − || z n || . By LaplaceTheorem, we have H ( t ) = r th ′′ (0)2 π Z + ∞− exp ( − th ( u )) du ∼ , t → + ∞ . Let η >
0. We have r th ′′ (0)2 π Z [ − , − η ] ∪ [ η, + ∞ [ exp ( − th ( u )) du = O ( e − c ( η ) t ) , t → + ∞ . INGULAR VALUES OF HANKEL OPERATORS 29
Let H + η ( t ) = r th ′′ (0)2 π Z
Let T be a compact operator between two complex Hilbert spaces. The decreasingsequence of singular values of T will be denoted by ( s n ( T )) n .The counting function of thesingular values of T is denoted by n ( s, T ) = { n : s n ( T ) ≥ s } , s > . Recall that C p denotes the class of increasing continuous function ψ : (0 , + ∞ ) → (0 , + ∞ )satisfying ψ ( αt ) ∼ α p ψ ( t ) , ( t → + ) , As before, D ψ ( T ) , d ψ ( T ) are given by D ψ ( T ) := lim sup s → + ψ ( s ) n ( s, T ) and d ψ ( T ) := lim inf s → + ψ ( s ) n ( s, T ) . The goal of this Annex is to extend the results obtained for ψ ( t ) = t p , [5, 25], to the class C p . We give here the proof for completeness. Proposition 8.1.
Let T and V be compact operators and λ ∈ C . We have(1) D ψ ( λT ) = | λ | p D ψ ( T ) and d ψ ( λT ) = | λ | p d ψ ( T ) .(2) D ψ ( T + V ) p +1 ≤ D ψ ( T ) p +1 + D ψ ( V ) p +1 .(3) d ψ ( T + V ) p +1 ≤ d ψ ( T ) p +1 + D ψ ( V ) p +1 .(4) D ψ ( T V ) ≤ D ψ ◦ t ( T ) D ψ ◦ t ( V ) .Proof. . For λ = 0, we have D ψ ( λT ) = lim sup s → + ψ ( s ) n ( s, λT ) = lim sup s → + ψ ( s ) n ( s/ | λ | , T )= lim sup s → + ψ ( | λ | s ) n ( s, T )= | λ | p lim sup s → + ψ ( s ) n ( s, T )= | λ | p D ψ ( T ) . Similarly, we have d ψ ( λT ) = | λ | p d ψ ( T ), ∀ λ ∈ C .2 . Let x ∈ (0 , s >
0, we have n ( s, T + V ) = n ( xs + (1 − x ) s, T + V ) ≤ n ( xs, T ) + n ((1 − x ) s, V )= n (cid:18) s, x T (cid:19) + n (cid:18) s, − x V (cid:19) . Hence, D ψ ( T + V ) ≤ D ψ (cid:0) x T (cid:1) + D ψ (cid:0) − x V (cid:1) . By the first assertion we obtain D ψ ( T + V ) ≤ x − p D ψ ( T ) + (1 − x ) − p D ψ ( V ) . It follows that D ψ ( T + V ) ≤ min x ∈ (0 , (cid:8) x − p D ψ ( T ) + (1 − x ) − p D ψ ( V ) (cid:9) , and 2 . is obtained.By the same way we obtain also the third assertion.4 . For all α, s >
0, we have n ( s, T V ) = n ( α √ s α − √ s, T V ) ≤ n ( α √ s, T ) + n ( α − √ s, V )= n ( √ s, α T ) + n ( √ s, αV ) . INGULAR VALUES OF HANKEL OPERATORS 31
Hence ψ ( s ) n ( s, T V ) ≤ ψ ( s ) n ( √ s, α T ) + ψ ( s ) n ( √ s, αV )= ˜ ψ ( √ s ) n ( √ s, α T ) + ˜ ψ ( √ s ) n ( √ s, αV ) , with ˜ ψ = ψ ( t ). Then D ψ ( T V ) ≤ D ˜ ψ (cid:18) α T (cid:19) + D ˜ ψ ( αV )= α − p D ˜ ψ ( T ) + α p D ˜ ψ ( V ) . It follows that D ψ ( T V ) ≤ min α> (cid:8) α − p D ˜ ψ ( T ) + α p D ˜ ψ ( V ) (cid:9) ≤ D ˜ ψ ( T ) D ˜ ψ ( V ) . (cid:3) Let us denote Σ ψ := { T ∈ S ∞ : ψ ( s ) n ( s, T ) = O (1) } and and Σ ψ := { T ∈ S ∞ : ψ ( s ) n ( s, T ) = o (1) } . The second and the third assertions of the Proposition 8.1 imply the following corollary.
Corollary 8.2.
Let T and V be compact operators. Suppose that V ∈ Σ ψ , then D ψ ( T + V ) = D ψ ( T ) and d ψ ( T + V ) = d ψ ( T ) . Once we have that, we can state the Theorem of A. Pushnitski (cid:2) [25], Theorem 2.2 (cid:3) inthe following form
Theorem 8.3.
Let T , T , ..., T n be a compact operators such that T ∗ i T j ∈ Σ ψ ◦√ . and T i T ∗ j ∈ Σ ψ ◦√ . , ∀ i = j. Then D ψ n X k =1 T k ! = lim sup s → + ψ ( s ) n X k =1 n ( s, T k ) and d ψ n X k =1 T k ! = lim inf s → + ψ ( s ) n X k =1 n ( s, T k ) . In particular, if for all s > , we have n ( s, T k ) = n ( s, T ) , k = 1 , , ..., n , then D ψ n X k =1 T k ! = nD ψ ( T ) and d ψ n X k =1 T k ! = nd ψ ( T ) . Proof.
Let T = n X k =1 T k . Since T T ∗ = n X i =1 T i T ∗ i + n X i,j =1 ,i = j T i T ∗ j , and T i T ∗ j ∈ Σ ψ ◦√ . , for i = j , we have D ψ ( T ) = D ψ ◦√ . ( T T ∗ ) = D ψ ◦√ . n X i =1 T i T ∗ i ! = D ψ ◦√ . (( J A )( J A ) ∗ )= D ψ ◦√ . (( J A ) ∗ ( J A )) . Where J : H n −→ H ( f , ..., f n ) −→ n X i =1 f i and A : H n −→ H n ( f , ..., f n ) −→ ( T f , T f , ..., T n f n )The matrix of ( J A ) ∗ ( J A ) is T ∗ T T ∗ T . . . T ∗ T n T ∗ T T ∗ T . . . T ∗ T n ... ... . . . ... T ∗ n T T ∗ n T . . . T ∗ n T n = T ∗ T T ∗ T . . .0 T ∗ n T n + O T ∗ T . . . T ∗ T n T ∗ T . . . T ∗ T n ... ... . . . ... T ∗ n T T ∗ n T . . . =: T ′ + T ′′ . Since T ∗ i T j ∈ Σ ψ ◦√ . , for i = j , we deduce that D ψ ( T ) = D ψ ◦√ . ( T ′ + T ′′ ) = D ψ ◦√ . ( T ′ )= D ψ ◦√ . ( T ∗ T , T ∗ T , ..., T ∗ n T n ) . Let now T the operator T := T T . . .0 T n We have D ψ ( T ) = D ψ ◦√ . ( T ∗ T , T ∗ T , ..., T ∗ n T n ) = D ψ ◦√ . ( T ∗ T )= D ψ ( T )= lim sup s → + ψ ( s ) n X k =1 n ( s, T k ) . INGULAR VALUES OF HANKEL OPERATORS 33
In the same way we prove that d ψ n X k =1 T k ! = lim inf s → + ψ ( s ) n X k =1 n ( s, T k ) . (cid:3) References
1. J. Arazy, S. D. Fisher, S. Janson, and J. Peetre,
Membership of Hankel operators on the ball in unitaryideals , J. Lond. Math. Soc. (1991), no. 3, 485–508.2. J. Arazy, S. D. Fisher, and J. Peetre, Hankel operators on weighted Bergman spaces , Amer. J. Math. (1988), no. 6, 989–1053.3. S. Axler,
The Bergman space, the Bloch space, and commutators of multiplication operators , DukeMath. J. (1986), no. 2, 315–332.4. B. Berndtsson, The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman ,Ann. Inst. Fourier (1996), no. 4, 1083–1094.5. M. Sh. Birman and M. Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space , 1 ed.,Mathematics and its Applications, vol. 5, Springer, 2012.6. H. Bommier-Hato, M. Engliˇs, and E. H. Youssfi,
Dixmier classes on generalized Segal–Bargmann–Fockspaces , J. Funct. Anal. (2014), no. 4, 2096 – 2124.7. A. Borichev, R. Dhuez, and K. Kellay,
Sampling and interpolation in large Bergman and Fock spaces ,J. Funct. Anal. (2007), no. 2, 563 – 606.8. M. R. Dostani´c,
Spectral properties of the commutator of Bergman’s projection and the operator ofmultiplication by an analytic function , Can. J. Math. (2004), no. 2, 277–292.9. O. El-Fallah and M. El Ibbaoui, Trace estimates of Toeplitz operators on Bergman spaces and appli-cations to Composition operators , preprint.10. ,
On the singular values of compact composition operators , Comptes Rendus Mathematique (2016), no. 11, 1087–1091.11. O. El-Fallah, H. Mahzouli, I. Marrhich, and H. Naqos,
Asymptotic behavior of eigenvalues of Toeplitzoperators on the weighted analytic spaces , J. Funct. Anal. (2016), no. 12, 4614 – 4630.12. ,
Toeplitz operators on harmonically weighted Bergman spaces and applications to compositionoperators on Dirichlet spaces , J. Math. Anal. Appl. (2018), no. 1, 471–489.13. M. Engliˇs and R. Rochberg,
The Dixmier trace of Hankel operators on the Bergman space , J. Funct.Anal. (2009), no. 5, 1445 – 1479.14. A. Erd´elyi,
Asymptotic expansions , int´egrale ed., Dover books on advanced mathematics, no. 3, DoverPublications, 1956.15. P. Galanopoulos and J. Pau,
Hankel operators on large weighted Bergman spaces , Ann. Acad. Sci.Fenn. Math. (2012), 635–648.16. Z. Hu, X. Lv, and A. P. Schuster, Bergman spaces with exponential weights , J. Funct. Anal. (2019),no. 5, 1402 – 1429.17. S. Janson,
Hankel operators between weighted Bergman spaces , Ark. Mat. (1988), no. 1-2, 205–219.18. P. Lin and R. Rochberg, Trace ideal criteria for Toeplitz and Hankel operators on the weighted Bergmanspaces with exponential type weights , Pacific J. Math. (1996), no. 1, 127–146.19. D. H. Luecking,
Trace ideal criteria for Toeplitz operators , J. Funct. Anal. (1987), no. 2, 345–368.20. D. H. Luecking, Characterizations of certain classes of Hankel operators on the Bergman spaces of theunit disk , J. Funct. Anal. (1992), no. 2, 247 – 271.21. J. Marzo and J. Ortega-Cerd`a,
Pointwise estimates for the Bergman kernel of the weighted Fock space ,J. Geom. Anal. (2009), no. 4, 890–910.
22. J. Pau,
Characterization of Schatten-class Hankel operators on weighted Bergman spaces , Duke Math.J. (2016), no. 14, 2771–2791.23. J. Pau, R. Zhao, and K. Zhu,
Weighted BMO and Hankel Operators Between Bergman Spaces , IndianaUniv. Math. J. (2016), no. 5, 1639–1673.24. V. Peller, Hankel operators and their applications , Springer Science & Business Media, 2012.25. A. Pushnitski,
Spectral asymptotics for Toeplitz operators and an application to banded matrices , TheDiversity and Beauty of Applied Operator Theory (Cham), Springer International Publishing, 2018,pp. 397–412.26. K. Seip and E. H. Youssfi,
Hankel operators on Fock spaces and related Bergman kernel estimates , J.Geom. Anal. (2013), no. 1, 170–201.27. R. Tytgat, Classe de Dixmier d’op´erateurs de Hankel , J. Operator Theory (2014), no. 1, 241–256.28. , Espace de Dixmier des op´erateurs de Hankel sur les espaces de Bergman `a poids , Czech. Math.J. (2015), no. 2, 399–426.29. K. Zhu, Operator theory in function spaces , second ed., Math. Surveys Monogr., vol. 138, Amer. Math.Soc, Providence, RI, 2007. MR 2311536
Mohammed V University in Rabat, Faculty of sciences, CeReMAR -LAMA- B.P. 1014Rabat, Morocco
Email address : [email protected] Mohammed V University in Rabat, Faculty of sciences, CeReMAR -LAMA- B.P. 1014Rabat, Morocco
Email address : [email protected]; [email protected] Laboratoire Mod´elisation, Analyse, Contrˆole et Statistiques, Faculty of sciences Ain-Chock, Hassan II University of Casablanca, B.P 5366 Maarif, Casablanca, Morocco
Email address : [email protected]; [email protected] Mohammed V University in Rabat, Faculty of sciences, CeReMAR -LAMA- B.P. 1014Rabat, Morocco
Email address ::