Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 160808, 15 pagesdoi:10.1155/2012/160808

Research Article

Derivatives of the Berezin Transform

Hélène Bommier-Hato

LATP, UMR CNRS 6632, CMI, Université de Provence, 39 rue Fjoliot-Curie, 13453 Marseille Cedex 13, France

Received 3 July 2009; Accepted 21 December 2011

Academic Editor: Miroslav Englis

Copyright © 2012 Hélène Bommier-Hato. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For a rotation invariant domain , we consider the Bergman space and we investigate some properties of the rank one projection . We prove that the trace of all the strong derivatives of A(z) is zero. We also focus on the generalized Fock space , where is the measure with weight , , with respect to the Lebesgue measure on and establish estimations of derivatives of the Berezin transform of a bounded operator T on .

1. Introduction and Statement of the Main Results

We consider a rotation invariant open set in and a positive rotation invariant measure on ; we suppose that has moments of every order. Let be the Hilbert space of square integrable complex-valued functions on and its subspace consisting of holomorphic elements. We assume that for each compact set there exists such that for all It is known that is a closed space of and possesses a reproducing kernel : we have for all and .

For a bounded linear operator on , the Berezin transform of is the function defined on by where is the normalized reproducing kernel The case of the Fock space, where and is the Gaussian measure, was considered by Coburn, Englis, and Zhang. Coburn [1] has shown that is a Lipschitz function. Namely, for , the constant 2 being sharp (see [2]).

Englis and Zhang [3] have shown that has bounded derivatives of all orders. Namely, for any multi-indices , , there exists a constant , depending on , , and only, such that where will stand for and similarly for . Recently, the author extended Coburn’s result to weighted Fock spaces, corresponding to endowed with the measure where is a positive parameter and is the normalized Lebesgue measure on , such that the volume of the unit ball in is equal to . We have showed that satisfies a local Lipschitz condition. There exist positive constants , , depending on and only, such that, for any with , there is a neighbourhood of that satisfies In this paper we will investigate some properties of the derivatives of the Berezin transform on . For in , if is the rank one projection we have We first fix some notations. Let denote the set of all -tuples with components in the set of all nonnegative integers. If , we let denote the length of and stands for . If satisfies for all , then we write . Our first main result is about the strong derivatives of .

Theorem 1.1. Let be a rotation invariant open set in and a rotation invariant positive measure on that satisfies (1.1) and has moments of every order. Moreover one assumes that, for any multi-index and any compact set of , there exists such that Then for all , multi-indices, the operators and are adjoint to each other; their rank is smaller or equal to the infimum of and . Moreover, if at least or is different from 0, one has

Our second main result generalizes the estimates of Englis and Zhang for the strong derivatives of the Berezin transform on weighted Fock spaces.

Theorem 1.2. For a bounded linear operator on , the Berezin transform has derivatives of all orders. In addition to any multi-indices , , there exist positive constants and , depending on , , and only such that

2. Preliminaries

We recall some properties of the Bergman kernel , when is a positive rotation invariant measure on . The kernel is given by for , in , with the usual convention that, for and , stands for and where By Lemma  2.1 in [4], we know that Thus we can write where is a holomorphic function of one complex variable.

For a bounded operator on , the Berezin transform can be written in the form where is the rank one projection .

Recall [3] that a mapping from a domain in into a Banach space possesses a strong holomorphic derivative at a point if similarly one can define the antiholomorphic derivative . Englis and Zhang showed that the mapping has strong derivatives.

Lemma 2.1. Let be a domain in , a measure on that satisfies (1.1). Then the function has strong derivatives of all orders.

Lemma 2.2. Under the same hypothesis, the trace-class-operator-valued function from into the space of trace-class operators on has strong derivatives of all orders.

The mapping and its derivatives can be expressed in terms of the function . It is easy tosee that, for , in ,

3. Proof of Theorem 1.1

We write, for and , in , Since it is possible to differentiate under the integral sign at any order with respect to , for in , we have The Leibnitz rule leads to Setting , we get and then where and . Notice that the rank of the operator is smaller than or equal to .

Setting now , , and for a fixed element in , we obtain, for in , Consequently, that is, It follows that the rank of the operator is also smaller than or equal to . Thus Now let and be multi-indices. Due to Lemma 2.2 and the continuity of the linear form , we have

4. Proof of Theorem 1.2

When and , for brevity we set for . The Bergman kernel of can be expressed in terms of the Mittag-Leffler function (see [5]). Putting , we have being the -th derivative of the Mittag-Leffler function , entire on and defined by In what follows, we will use some asymptotic properties of this function (see [6]) near the positive real axis.

Lemma 4.1. Let a nonnegative integer. There exists a constant , such that, for any complex number with , one has where and , for integer. The polynomials are defined by , is of degree , and when , for any nonnegative integers and .

We also need asymptotic estimates for some auxiliaries functions.

Lemma 4.2. Let and be nonnegative integers. When is real and tends to ,

For a bounded linear operator on , the Berezin transform is given by (see [3]) Let and be some multi-indices. By differentiation, Lemma 2.2 gives Due to the continuity of the bilinear form , we have for all bounded operators and trace class operators . Therefore, we obtain Like in Section 3, we set . Next we shall estimate .

Fix in and in . We recall that where and . Then To estimate the eigenvalues of the finite rank positive operator , we compute its trace: With the operator being of finite rank, we observe that . We now compute and then estimate each diagonal term . Fix some multi-indices and such that and . To simplify the notation we set and instead of and . We obtain For the second term, since we get By the Leibnitz rule, we see that Then Thus Setting and , we have where Setting for a multiindex , we see that Using the Leibnitz rule An induction process shows that Therefore Now let . It follows that On the other hand Thus, if we set , then Using the identity we see that Notice that, for and integers, we have, when is real and , Hence by Lemmas 4.1 and 4.2 we have the following estimates: for some constant when the real tends to . Thus for any multiindex , there exists a constant such that . Then taking the sum over we get Since and the operator is positive, its eigenvalues are bounded above by . Thus the eigenvalues of are bounded above by . Since is of finite rank, the proof is complete.

Acknowledgments

The author wishes to thank El Hassan Youssfi and Miroslav Englis for useful discussions.

References

L. A. Coburn, “A Lipschitz estimate for Berezin's operator calculus,” Proceedings of the American Mathematical Society, vol. 133, no. 1, pp. 127–131, 2005.

L. A. Coburn, “Sharp Berezin Lipschitz estimates,” Proceedings of the American Mathematical Society, vol. 135, no. 4, pp. 1163–1168, 2007.

M. Englis and G. Zhang, “On the derivatives of the Berezin transform,” Proceedings of the American Mathematical Society, vol. 134, no. 8, pp. 2285–2294, 2006.

S. Lovera and E. H. Youssfi, “Spectral properties of the -canonical solution operator,” Journal of Functional Analysis, vol. 208, no. 2, pp. 360–376, 2004.

H. Bommier-Hato and E. H. Youssfi, “Hankel operators on weighted Fock spaces,” Integral Equations and Operator Theory, vol. 59, no. 1, pp. 1–17, 2007.

H. Bommier-Hato, “Lipschitz estimates for the Berezin transform,” Journal of Function Spaces and Applications, vol. 8, no. 2, pp. 103–128, 2010.