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Some Hilbert Spaces Related with the Dirichlet Space Some Hilbert spaces related with the Dirichlet space Article Published Version Creative Commons: Attribution-Noncommercial-No Derivative Works 4.0 Open access Arcozzi, N., Mozolyako, P., Perfekt, K.-M., Richter, S. and Sarfatti, G. (2016) Some Hilbert spaces related with the Dirichlet space. Concrete Operators, 3 (1). 0011. ISSN 2299- 3282 doi: https://doi.org/10.1515/conop-2016-0011 Available at http://centaur.reading.ac.uk/71561/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . Published version at: http://dx.doi.org/10.1515/conop-2016-0011 To link to this article DOI: http://dx.doi.org/10.1515/conop-2016-0011 Publisher: De Gruyter All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online Concr. Oper. 2016; 3: 94–101 Concrete Operators Open Access Research Article Nicola Arcozzi*, Pavel Mozolyako, Karl-Mikael Perfekt, Stefan Richter, and Giulia Sarfatti Some Hilbert spaces related with the Dirichlet space DOI 10.1515/conop-2016-0011 Received December 23, 2015; accepted May 16, 2016. Abstract: We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space. Keywords: Dirichlet space, Complete Nevanlinna Property, Hilbert-Schmidt operators, Carleson measures MSC: 30H25, 47B35 1 Introduction Consider the Dirichlet space D on the unit disc z C z < 1 of the complex plane. It can be defined as the f 2 W j j g Reproducing Kernel Hilbert Space (RKHS) having kernel n 1 1 X1 .zw/ kz.w/ k.w; z/ log : D D zw 1 zw D n 1 n 0 D C d We are interested in the spaces Dd having kernel k , with d N. Dd can be thought of in terms of function spaces 2 d on polydiscs, following ideas of Aronszajn [4]. To explain this point of view, note that the tensor d-power D˝ d of the Dirichlet space has reproducing kernel kd .z1; ; zd w1; : : : ; wd / …j 1k.zj ; wj /. Hence, the space d I D D d of restrictions of functions in D to the diagonal z1 zd has the reproducing kernel k , and therefore ˝ D D coincides with Dd . We will provide several equivalent norms for the spaces Dd and their dual spaces in Theorem 1.1. Then we will discuss the properties of these spaces. More precisely, we will investigate: – Dd and its dual space HSd in connection with Hankel operators of Hilbert-Schmidt class on the Dirichlet space D; – the complete Nevanlinna-Pick property for Dd ; – the Carleson measures for these spaces. Concerning the first item, the connection with Hilbert-Schmidt Hankel operators served as our original motivation for studying the spaces Dd . *Corresponding Author: Nicola Arcozzi: Università di Bologna, Dipartimento di Matematica, Piazza di Porta S.Donato 5, Bologna, E-mail: [email protected] Pavel Mozolyako: Chebyshev Lab at St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia, E-mail: [email protected] Karl-Mikael Perfekt: Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway, E-mail: [email protected] Stefan Richter: Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA, E-mail: [email protected] Giulia Sarfatti: Istituto Nazionale di Alta Matematica “F. Severi”, Città Universitaria, Piazzale Aldo Moro 5, 00185 Roma, and Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4, place Jussieu, F-75252 Paris, France, E-mail: [email protected] © 2016 Arcozzi et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.Brought to you by | University of Reading Authenticated Download Date | 7/24/17 4:58 PM Some Hilbert spaces related with the Dirichlet space 95 Note that the spaces D live infinitely close to D in the scale of weighted Dirichlet spaces D , defined by the d Q s norms ZC 2 Z 2 ˇ it ˇ dt ˇ ˇ2 2 s dA.z/ ' ˇ'.e /ˇ ˇ'0.z/ˇ .1 z / ; 0 s < 1; Ds ˇ ˇ k k Q D 2 C j j Ä z <1 j j dA.z/ where is normalized area measure on the unit disc. d Notation: We use multiindex notation. If n .n1; : : : ; nd / belongs to N , then n n1 nd . We write D j j D C C A B if A and B are quantities that depend on a certain family of variables, and there exist independent constants 0 < c < C such that cA B CA. Ä Ä Equivalent norms for the spaces Dd and their dual spaces HSd Theorem 1.1. Let d be a positive integer and let X 1 ad .k/ : D .n1 1/ : : : .nd 1/ n k j jD C C P k Then the norm of a function '.z/ k1 0 b'.k/z in Dd is D D !1=2 X1 1 2 ' D ad .k/ '.k/ Œ'd ; (1) k k d D jb j k 0 D where !1=2 X1 k 1 2 Œ'd C '.k/ : (2) D d 1 jb j k 0 log .k 2/ D C An equivalent Hilbert norm Œ' d Œ'd for ' in terms of the values of ' is given by j j 0 11=2 Z 2 2 1 dA.z/ Œ' d '.0/ @ '0.z/ A : (3) j j D j j C j j d 1 1 Á log 1 z 2 D j j Define now the holomorphic space HSd by the norm: !1=2 ˇ ˇ2 X1 2 ˇ ˇ HSd .k 1/ ad .k/ b.k/ : (4) k k D C ˇ ˇ k 0 D Then, HSd .Dd / is the dual space of Dd under the duality pairing of D. Moreover, Á !1=2 ˇ ˇ2 X1 d 1 ˇ ˇ HSd Œ HSd .k 1/ log .k 2/ b.k/ k k WD C C ˇ ˇ k 0 0 D 11=2 Z  à 2 2 d 1 1 dA.z/ Œ HSd @ .0/ 0.z/ log A : (5) j j WD j j C j j 1 z 2 D j j Furthermore, the norm can be written as 2 X 2 en : : : en ; D ; (6) k kHSd D jh 1 d i j .n1;:::;nd / zn where en is the canonical orthonormal basis of D, en.z/ . n1 0 pn 1 f g D D C Brought to you by | University of Reading Authenticated Download Date | 7/24/17 4:58 PM 96 N. Arcozzi et al. The remainder of this section is devoted to the proof of Theorem 1.1. The expression for ' Dd in (1) follows d k k by expanding .kz/ as a power series. The equivalence ' D Œ'd , as well as ' HS Œ'HS , are k k d k k d d consequences of the following lemma. We denote by c; C positive constants which are allowed to depend on d only, whose precise value can change from line to line. Lemma 1.2. For each d N there are constants c; C > 0 such that for all k 0 we have 2 d 1 log .k 2/ cad .k/ C Cad .k/: Ä k 1 Ä C Consequently, if t .0; 1/, then 2  Ãd d 1  Ãd 1 1 X1 log .k 2/ 1 1 c log C t k C log : t 1 t Ä k 1 Ä t 1 t k 0 D C Proof of Lemma 1.2. We will prove the Lemma by induction on d N. It is obvious for d 1. Thus let d 2 2 D and suppose the lemma is true for d 1. Also we observe that there is a constant c > 0 such that for all k 0 and 0 n k we have Ä Ä c logd 2.k 2/ logd 2.n 2/ logd 2.k n 2/ 2 logd 2.k 2/: C Ä C C C Ä C Then for k 0 X 1 ad .k/ D .n1 1/ : : : .nd 1/ n1 nd k CC D C C k X 1 X 1 D n 1 .n2 1/ : : : .nd 1/ n 0 n2 nd k n D C CC D C C k d 2 X 1 log .k n 2/ C by the inductive assumption n 1 k n 1 n 0 D C C k d 2 d 2 1 X log .n 2/ log .k n 2/ C C C D 2 .n 1/.k n 1/ n 0 D C C k X 1 logd 2.k 2/ by the earlier observation C .n 1/.k n 1/ n 0 D C C d 2 k log .k 2/ X 1 1 C D k 2 n 1 C k n 1 n 0 C D C C logd 1.k 2/ C : k 1 C Next, we prove the equivalence Œ'HS Œ' HS which appears in (5). d j j d Lemma 1.3. Let d N. Then 2 1  Ãd 1 d 1 Z 1 1 log .k 2/ t k log dt C ; k d: t 1 t k 1 0 C Given the Lemma, we expand Z ˇ ˇ2 2 2 ˇ X1 k 1ˇ d 1 1 dA.z/ Œ b.0/ ˇ b.k/kz ˇ log j jHSd D j j C ˇ ˇ 1 z 2 ˇk 1 ˇ D D j j 1 2 Z 2 X1 2 ˇ ˇ d 1 1 k 1 b.0/ k ˇ b.k/ˇ log t dt D j j C ˇ ˇ 1 t k 1 D 0 Brought to you by | University of Reading Authenticated Download Date | 7/24/17 4:58 PM Some Hilbert spaces related with the Dirichlet space 97 2 d 1 2 X1 2 ˇ ˇ log .k 2/ b.0/ k ˇ b.k/ˇ C j j C ˇ ˇ k 1 k 1 C Œ 2 ; D HSd obtaining the desired conclusion.
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