Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Mathematics Department, Middle East Technical University, Ankara 06531, Turkey E-mail: dany@math.bgu.ac.il

Tam metin

(1)C. R. Acad. Sci. Paris, t. 333, Série I, p. 285–290, 2001 Analyse mathématique/Mathematical Analysis. Integral formulas for a sub-Hardy Hilbert space on the ball with complete Nevanlinna–Pick reproducing kernel b ˘ Daniel ALPAY a , H. Turgay KAPTANOGLU a b. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Mathematics Department, Middle East Technical University, Ankara 06531, Turkey E-mail: dany@math.bgu.ac.il; kaptan@math.metu.edu.tr. (Reçu le 8 mai 2001, accepté le 11 juin 2001). Abstract.. We use the theory of Hilbert spaces of analytic functions on bounded symmetric domains in CN to obtain information on the (1/(N + 1))st power of the Bergman kernel of the ball. This kernel has played recently an important and growing role in operator theory. We present several integral formulas for the Hilbert space generated by this kernel.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Représentation intégrale pour un sous-espace de Hardy de la boule de noyau reproduisant de type Nevanlinna–Pick Résumé.. Nous utilisons la théorie des espaces de Hilbert de fonctions analytiques dans des domaines symétriques de CN pour étudier la puissance 1/(N + 1) du noyau de Bergman de la boule unité. Ce noyau joue depuis peu un rôle important en théorie des opérateurs. Nous donnons des représentations intégrales pour l’espace à noyau reproduisant associé.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Version française abrégée Soit H l’espace à noyau reproduisant de fonctions analytiques dans la boule unité B de CN de noyau reproduisant k(z, w) = (1−(z1 w 1 +· · ·+zN wN ))−1 . Les monômes {z α } forment un ensemble orthogonal dense de H, α 2 α! z = α! et (f, g)H = , fα g α H |α|! |α|! α où f (z) = α fα z α et g(z) = α gα z α sont dans H, et où nous utilisons les indices multiples. Cet espace joue un rôle important depuis peu en théorie des opérateurs, et en particulier en relation avec le problème d’interpolation de Nevanlinna–Pick. L’espace H est inclus contractivement dans l’espace de Hardy. L’inclusion est stricte dès que N > 1. Nous avons étudié certains problèmes d’interpolation dans la note [2]. Cette Note est consacrée à des interprétations géométriques du produit scalaire de H. Soit σ la mesure d’aire sur ∂B, normalisée par σ(∂B) = 1, et soit ν la mesure de volume dans B avec Note présentée par Jean-Pierre K AHANE. S0764-4442(01)02046-8/FLA  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 285.

(2) D. Alpay, H.T. Kaptano˘glu. la normalisation ν(B) = 1. Nous dénotons par τ la mesure dτ (z) = k(z, w)N +1 dν(z). Cette mesure est invariante sous l’action des automorphismes de B. Si f est une fonction analytique, le symbole Rf (z) désigne sa dérivée radiale au point z définie par (4). Soit I l’opérateur d’identité et pour des entiers m, j, soit Rm = (R + (N − 1)I)(R + (N − 2)I) · · · (R + (N − m)I), Dm = (R + I)(R + 2I) · · · (R + mI) et Sj = RN −1 (R + jI). Les formules intégrales pour le produit scalaire de H sont données dans les théorèmes 2, 4, 5, 7, 9 et 10. Cela nous amène à des représentations intégrales pour les fonctions de H que nous présentons dans les corollaires 3, 6, 8 et 11. Nous obtenons aussi des représentations intégrales en termes de mesures planaires pour l’espace de Hardy.. 1. Sub-Hardy Hilbert space H with reproducing kernel (1 − z, w )−1 Let B be the open unit ball in CN and S its boundary, the unit sphere. The space H is the reproducing kernel Hilbert space on B with reproducing kernel k(z, w) = (1 − z, w )−1 , where z, w = z1 w1 + · · · + zN w N . This means that for any f ∈ H and w ∈ B, we have f (w) = (f, k(·, w))H , where (·, ·)H is the inner product of H. The functions in H are holomorphic in B. It is shown in [3] that the monomials {z α } form an orthogonal dense subset of H, and that α 2 α! z = α! and (f, g)H = , (1) fα g α H |α|! |α|! α if f (z) = α fα z α and g(z) = α gα z α are in H. Here and throughout we use multi-index notation; so α = (α1 , . . . , αN ) is an N -tuple of nonnegative integers, |α| = α1 + · · ·+ αN , α! = α1 ! · · · αN !, 00 = 1, and αN z α = z1α1 · · · zN . We adhere to the convention that an empty product is 1, and for operators is I, the identity operator. In addition, ( ) means the closure of a set or the complex conjugate. The space H has attracted a lot of attention lately in operator theory because of its importance in Nevanlinna–Pick interpolation; see [1] and [7]. The reproducing kernels of the Bergman space A2 and the Hardy space H2 of the ball are K(z, w) = k(z, w)N +1 and C(z, w) = k(z, w)N . We have H ⊂ H2 ⊂ A2 . All inclusions are contractive and proper as partly shown in [3], except that H and H2 coincide when N = 1.. 2. Preliminaries Let Aut(B) be the group of all automorphisms of B, that is, one-to-one holomorphic maps of B onto B. We refer the reader to [3] and [10, Chapter 2] for detailed descriptions of members ψ of Aut(B). We just note that ψ −1 = ψ and Aut(B) acts transitively on B. Let σ be the Lebesgue (surface) measure on S normalized so that σ(S) = 1. Let V be the Lebesgue (volume) measure on B, and let ν be its normalized form with ν(B) = 1. Let τ be the measure defined on B by dτ (z) = k(z, z)N +1 dν(z). The importance of τ lies in the fact that it is invariant under the automorphisms of B; that is, B f dτ = B (f ◦ ψ) dτ β for ψ ∈ Aut(B) and integrable f . Propositions 1.4.8 and 1.4.9 of [10] state that S ζ α ζ dσ(ζ) = 0 = α β B z z dν(z) if α = β, (N − 1)! α! N ! α! α 2 and . (2) |ζ | dσ(ζ) = |z α |2 dν(z) = (N − 1 + |α|)! (N + |α|)! S B The polar-coordinate formula that relates ν and σ given in 1.4.3 of [10] is 1 f dν = 2N r2N −1 dr f (rζ) dσ(ζ). B. 286. 0. S. (3).

(3) Integral formulas for a Nevanlinna–Pick reproducing kernel Hilbert space. L EMMA 1. – For Re λ > N , if α = β, then B z α z β (1 − |z|2 )λ dτ (z) = 0, and N! α! α 2 2 λ |z | (1 − |z| ) dτ (z) = . (λ − N ) · · · (λ − 2)(λ − 1) (λ)(λ + 1) · · · (λ + |α| − 1) B The lemma is proved by applying (3) to the integral on the left and then using the elementary properties of the beta function. The integral is with respect to the volume measure ν if the power on 1 − |z|2 is written as λ − N − 1. The second formula in (2) is the case λ = N + 1. The radial derivative at z of a holomorphic function f and its most useful applications are Rf (z) =. N j=1. zj. ∂f (z), ∂zj. R z α = |α| z α. . and R k(z, w)m = m z, w k(z, w)m+1 .. (4). The presentation below mainly originates in work on analysis on bounded symmetric domains that depend heavily on Lie-group machinery. Readable accounts are [6] and [5]. What is interesting is that the space H and its kernel k(z, w) is the special case λ = 1 in B of a whole family of spaces and kernels on bounded symmetric domains indexed by λ, and the range of values of λ, called the Wallach set, depends on the type of the domain Ω and its parameters such as genus g. The Wallach set W (Ω) is the set of all λ for which Kλ (z, w) = KΩ (z, w)λ/g is nonnegative definite, where KΩ is the Bergman kernel of Ω. Then reproducing kernel Hilbert spaces Hλ (Ω) are defined on Ω that admit Kλ (z, w) as reproducing kernels. These spaces include the Bergman and Hardy spaces as special cases. It would be interesting to investigate Nevanlinna–Pick-type and operator-theoretic problems on such spaces. There is a lot of room even in the unit ball, the simplest bounded symmetric domain, where g = N + 1 and W (B) = [0, ∞). So we have a Hilbert space Hλ on B with reproducing kernel Kλ (z, w) = (1 − z, w )−λ for each λ 0. Each of the unitary operators (Uψ f )(z) = f (ψ(z))[(Jψ)(z)]1/(N +1) maps H onto itself, where ψ ∈ Aut(B), (Jψ)(z) is its complex Jacobian, and the principal branch is used for the root. In fact, H is the unique Hilbert space of holomorphic functions on B with this property in which point evaluations are continuous. Further, the inner product of H is invariant under each Uψ , that is, (f ◦ Uψ , g ◦ Uψ )H = (f, g)H for f, g ∈ H. These are in [6]. 3. Integral representation with surface measure Let R0 = I and Rm = (R + (N − 1)I)(R + (N − 2)I) · · · (R + (N − m)I) for m = 1, 2, . . . . These linear differential operators are introduced in [4]. We need only the cases m = N − 1 and m = N . T HEOREM 2. – For f, g ∈ H, we have (f, g)H = lim− r→1. 1 (N − 1)!. (RN −1 f )(rζ) g(rζ) dσ(ζ). S. This theorem is an adaptation of a result in [4] to functions holomorphic in B, but not in a neighborhood of B. It suffices to check the formula on the monomials z α . This is an easy matter by (4) and (2). Since the monomials are dense in H, the Hilbert space completion of {z α } is just H. Let’s give some applications of this formula. By the reproducing property of the kernel k(z, w), if f ∈ H and w ∈ B, we have 1 (RN −1 k)(ζ, w) f (ζ) dσ(ζ), (5) f (w) = (k(·, w), f )H = (N − 1)! S where no limit as r → 1− is needed, because k(z, w) is holomorphic in a neighborhood of B, and f has boundary values almost everywhere since H ⊂ H2 . Now suppose f (0) is real, put u = Re f , define h(z, w) = 2k(z, w) − 1 =. 1 + z, w , 1 − z, w. 287.

(4) D. Alpay, H.T. Kaptano˘glu. and consider the pairing (u, h(·, w))H . This pairing does not make sense as it stands since u is not in H unless it is constant. However, it does make sense when the integral form of the inner product is used. Then . 1 u, h(·, w) H = (RN −1 h)(ζ, w) u(ζ) dσ(ζ) (N − 1)! S . 1 2(RN −1 k)(ζ, w) − (N − 1)! f (ζ) + f (ζ) dσ(ζ) = 2(N − 1)! S = f (w) + f (0) − Re f (0) = f (w).. (6). Above, f (w) follows from (5); the product of (RN −1 k)(ζ, w) and f (ζ) give f (0) as seen by expanding them into power series and using (2); the mean-value property of the harmonic function u yields Re f (0). Thus we obtain a Herglotz-type representation with a Herglotz-type kernel, and the functions in H are recovered from the boundary values of their real parts using the integral (6). C OROLLARY 3. – For f ∈ H, we have. . f (w) = S. f (ζ) dσ(ζ). (1 − w, ζ )N. This follows by computing the radial derivatives in (5) explicitly using (4). Thus we obtain the Cauchy– Szegö representation in B [10, 3.2.4]. This is not surprising because of the last paragraph of Section 1. Correspondingly, the explicit version of (6) is the Herglotz formula in B [10, 3.2.5]. 4. Integral representations with volume measure T HEOREM 4. – For f, g ∈ H, we have 1 (f, g)H = f (0)g(0) + N!. (RN f )(z) g(z) B. 1 dν(z). |z|2N. This formula also appears in [4]. Again we check only the norms of the monomials using (4) and (3). Other families of linear differential operators on bounded symmetric domains are introduced in [11], but explicit expressions for most of them apparently are not known in general; see [5, p. 55]. However, in B only one such operator is needed and it is computed in [5, p. 55] for a different kind of domain; yet it works in B too. It turns out that these operators are also relatives of the radial derivative in B. Put D0 = I and Dm = (R + I)(R + 2I) · · · (R + mI) for m = 1, 2, . . .. T HEOREM 5. – For f, g ∈ H and for any integer m N , we have m 1 (f, g)H = D f (z) g(z) k(z, z)N −m dν(z). N ! (m − N )! B As usual, it is sufficient to check only the monomials. This is done by applying (4) and then Lemma 1 with λ = m + 1. Note that the power on k(z, z) is never positive. C OROLLARY 6. – For integers m N and f ∈ H, we have

(5) m (1 − |z|2 )m−N f (w) = f (z) dν(z). m+1 N B (1 − w, z ) The proof is similar to that of Corollary 3 using the operators Dm in place of RN −1 . What we obtain now are those cases with integer powers of the Bergman-type projections discussed in [10, Section 7.1]. The case m = N , in particular, is the Bergman projection. This is not surprising either because of the last paragraph of Section 1. The corresponding Herglotz-type formulas look complicated and are not given.. 288.

(6) Integral formulas for a Nevanlinna–Pick reproducing kernel Hilbert space. We now introduce some new integral formulas for the inner product of H that are intermediate between those in Theorems 4 and 5. First let Sj = RN −j (R + jI)(R + (j − 1)I) · · · (R + I) = RN −1 (R + jI) for j = 1, . . . , N so that SN = DN . Let also S0 = RN . T HEOREM 7. – For f, g ∈ H and for j = 1, 2, . . . , N , we have 1 1 (f, g)H = (Sj f )(z) g(z) 2(N −j) dν(z). N! B |z| The formula holds also for j = 0 provided we add to the integral the term f (0)g(0). The proof of this follows the same lines as that of Theorem 4. Clearly, the case j = 0 is in Theorem 4, and the case j = N is the case m = N of Theorem 5. The case j = 1 also appears in [4]. C OROLLARY 8. – For j = 1, . . . , N and f ∈ H, we have 1 j + (N − j) w, z f (w) = f (z) dν(z). N B (1 − w, z )N +1 |z|2(N −j) The formula holds also for j = 0 provided we add to the integral the term f (0). We follow the proof of Corollary 3 again using the operators Sj in place of RN −1 . The case j = N gives the Bergman formula again. The only simple-looking Herglotz-type formula is that of j = 0 and is 2 w, z u(z) dν(z). f (w) = f (0) + N +1 |z|2N B (1 − w, z ) 5. Integral representations with invariant measure Now we obtain a new and simpler integral formula that describes H as a weighted-Bergman-type space. Following the method of [9], consider the pairing (λ − N ) · · · (λ − 2)(λ − 1). f, g λ = f (z) g(z) k(z, z)−λ dτ (z). N! B T HEOREM 9. – For f, g ∈ H, we have (f, g)H = limλ→1. f, g λ , and H is the space of all holomorphic functions f on B that satisfy limλ→1. f, f λ < ∞. The advantage of this formula is that there is no derivative on either function. The weight in the integral is essentially (1 − |z|2 )−N with respect to the measure dν(z). Proof. – We start with the monomials once again. If λ > N , then by Lemma 1,. z α , z α λ =. α! . (λ)(λ + 1) · · · (λ + |α| − 1). Thus the function λ →. z α , z α λ extends holomorphically at least to the region {Re λ > 0}. Moreover, limλ→1. z α , z α λ = α!/|α|!. It follows that the Hilbert space completion of {z α } coincides with H. ✷ We again introduce some new integral formulas for the inner product of H that are intermediate this time between those in Theorems 5 and 9. T HEOREM 10. – For f, g ∈ H and m = 0, 1, . . . , N , we have (λ − N + m) · · · (λ − 2)(λ − 1) m (f, g)H = lim D f (z) g(z) k(z, z)−(m+λ) dτ (z). λ→1 N! B The proof of this follows the same lines as that of Theorem 9. Clearly, the case m = 0 is Theorem 9, and the case m = N is the case m = N of Theorem 5. In the latter case, no limit is actually required and we can set λ = 1 directly.. 289.

(7) D. Alpay, H.T. Kaptano˘glu. C OROLLARY 11. – For m = 0, 1, . . . , N and f ∈ H, we have m! (λ − N + m) · · · (λ − 2)(λ − 1) (1 − |z|2 )m−N +λ−1 f (z) dν(z). f (w) = lim λ→1 N! (1 − w, z )m+1 B The proof is as above. The formulas obtained are the extensions of the Bergman-type projections in [10], Section 7.1 to certain negative integer values of m − N for the space H (see Corollary 6). The case m = N gives the Bergman integral once again and requires no limit. The Herglotz-type formula for m = 0 is (λ − N ) · · · (λ − 2)(λ − 1) 1 + w, z f (w) = lim (1 − |z|2 )λ u(z) dτ (z). λ→1 N! B 1 − w, z 6. Hardy space revisited The inner products in Theorems 4, 5, 7, 9 and 10 written explicitly when N = 1 give new and unusual formulas for the inner product of H2 . Here is a short list. 1 2π 1 iθ f r e g(r eiθ ) eiθ dr dθ (f, g)H2 = f (0)g(0) + π 0 0 . 1 2π 1 = f (r eiθ ) + f (r eiθ ) r eiθ g r eiθ r dr dθ π 0 0 λ−2 λ − 1 2π 1 iθ iθ = lim 1 − r2 f re g re r dr dθ. λ→1 π 0 0 The last formula is actually identical to the classical form of the inner product of H2 , because the weak*limit as λ → 1 of the measures dµλ (r) = (λ − 1) (1 − r2 )λ−2 2r dr on [0, 1] is the unit point mass at r = 1. By Theorem 2.3 of [8], this is seen by computing the pointwise limit as λ → 1 of their distribution functions, which are Fλ (x) = µλ ([0, x]) = 1 − (1 − x2 )λ−1 for 0 x < 1 and Fλ (1) = µλ ([0, 1]) = 1. Acknowledgements. The research of the first author was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and by the Israeli Academy of Sciences.. References [1] Agler J., McCarthy J.E., Complete Nevanlinna–Pick kernels, J. Funct. Anal. 175 (2000) 111–124. [2] Alpay D., Kaptano˘glu H.T., Sous espaces de codimension finie dans la boule unité et un problème de factorisation, C. R. Acad. Sci. Paris, Série I 331 (2000) 947–952. [3] Alpay D., Kaptano˘glu H.T., Some finite-dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem, Integral Equations Operator Theory (to appear). [4] Arazy J., Integral formulas for the invariant inner products in spaces of analytic functions on the unit ball, in: Function Spaces, Lecture Notes in Pure and Appl. Math., Vol. 136, Dekker, New York, 1992, pp. 9–23. [5] Arazy J., A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains, in: Contemp. Math., Vol. 185, Amer. Math. Soc., Providence, 1995, pp. 7–65. [6] Arazy J., Fisher S.D., Invariant Hilbert spaces of analytic functions on bounded symmetric domains, in: Topics in Operator Theory, Oper. Theory Adv. Appl., Vol. 48, Birkhäuser, Basel, 1990, pp. 67–91. [7] Ball J.A., Trent T., Vinnikov V., Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, in: Oper. Theory Adv. Appl., Vol. 122, Birkhäuser, Basel, 2001, pp. 89–138. [8] Billingsley P., Weak Convergence of Measures: Applications in Probability, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, 1971. [9] Inoue T., Automorphism invariant inner product in Hilbert spaces of holomorphic functions on the unit ball of Cn , Osaka J. Math. 32 (1995) 227–236. [10] Rudin W., Function Theory in the Unit Ball of Cn , Grundlehren Math. Wiss., Vol. 241, Springer, New York, 1980. [11] Yan Z., Invariant differential operators and holomorphic function spaces, J. Lie Theory 10 (2000) 1–31.. 290.

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