Graduate Texts in Mathematics 199 Editorial Board S Axler F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics T AKEUTriZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nded HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAc LANE Categories for the Working Mathematician 2nd ed HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTriZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory 10 COHEN A Course in Simple Homotopy Theory II CONWAY Functions of One Complex Variable I 2nd ed 12 BEALS Advanced Mathematical Analysis 13 ANDERSON/FULLER Rings and Categories of Modules 2nd ed 14 GOLUBITSKy/GUlLLEMIN Stable Mappings and Their Singularities 15 BERBERIAN Lectures in Functional Analysis and Operator Theory 16 WINTER The Structure of Fields 17 ROSENBLATT Random Processes 2nd ed 18 HALMOS Measure Theory 19 HALMOS A Hilbert Space Problem Book 2nded 20 HUSEMOLLER Fibre Bundles 3rd ed 21 HUMPHREYS Linear Algebraic Groups 22 BARNES/MACK An Algebraic Introduction to Mathematical Logic 23 GREUB Linear Algebra 4th ed 24 HOLMES Geometric Functional Analysis and Its Applications 25 HEWITT/STROMBERG Real and Abstract Analysis 26 MANES Algebraic Theories 27 KELLEY General Topology 28 ZARlSKriSAMUEL Commutative Algebra Vol.I 29 ZARlSKriSAMUEL Commutative Algebra Vol.lI 30 JACOBSON Lectures in Abstract Algebra I Basic Concepts 31 JACOBSON Lectures in Abstract Algebra II Linear Algebra 32 JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 HIRSCH Differential Topology SPITZER Principles of Random Walk 2nded ALEXANDERiWERMER Several Complex Variables and Banach Algebras 3rd ed KELLEy/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERTIFRITZSCHE Several Complex Variables ARVESON An Invitation to C*-Algebras KEMENY/SNELLiKNAPP Denumerable Markov Chains 2nd ed ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEVE Probability Theory I 4th ed LOEVE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and SACHS/WU General Relativity for Mathematicians GRUENBERG/WEIR Linear Geometry 2nd ed EDWARDS Fermat's Last Theorem KLINGENBERG A Course in Differential Geometry HARTSHORNE Algebraic Geometry MANIN A Course in Mathematical Logic GRAVERiW ATKINS Combinatorics with Emphasis on the Theory of Graphs BROWNIPEARCY Introduction to Operator Theory I: Elements of Functional Analysis MASSEY Algebraic Topology: An Introduction CROWELLlFox Introduction to Knot Theory KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed LANG Cyclotomic Fields ARNOLD Mathematical Methods in Classical Mechanics 2nd ed WHITEHEAD Elements of Homotopy (continued after index) Haakan Hedenmalm Boris Korenblum Kehe Zhu Theory of Bergman Spaces With Illustrations Springer Haakan Hedenmalm Department of Mathematics Lund University Lund, S-22100 Sweden Boris Korenblum Kehe Zhu Department of Mathematics State University of New York at Albany Albany, NY 12222-0001 USA Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department Bast Hall University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 47-01, 47A15, 32A30 Library of Congress Cataloging-in-Publication Data Hedenmalm, Haakan Theory of Bergman spaces I Haakan Hedenmalrn, Boris Korenblurn, Kehe Zhu p cm - (Graduate texts in rnathernatics ; 199) Includes bibliographical references and index ISBN 978-1-4612-6789-8 ISBN 978-1-4612-0497-8 (eBook) DOI 10.1007/978-1-4612-0497-8 Bergman kernel functions IV Series QA33l H36 2000 5l5-dc2l I Korenblurn, Boris 11 Zhu, Kehe, 1961- III Title 99-053568 Printed on acid-free paper © 2000 Springer Science+Business Media New York Originally published by Springer-Verlag New York Berlin Heidelberg in 2000 Softcover reprint of the hardcover 1st edition 2000 All rights reserved This work rnay not be translated or copied in whole or in part without the written permission of the Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar rnethodology now known or hereafter developed is forbidden The use of general descriptive narnes, trade narnes, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production rnanaged by Jenny Wolkowicki; rnanufacturing supervised by Jeffrey Taub Photocornposed copy prepared frorn the authors' LaTeX files 543 ISBN 978-1-4612-6789-8 Preface Their memorials are covered by sand, their rooms are forgotten But their names live on by the books they wrote, for they are beautiful (Egyptian poem, 1500 1000 BC) The theory of Bergman spaces experienced three main phases of development during the last three decades The early 1970's marked the beginning of function theoretic studies in these spaces Substantial progress was made by Horowitz and Korenblum, among others, in the areas of zero sets, cyclic vectors, and invariant subspaces An influential presentation of the situation up to the mid 1970 's was Shields' survey paper "Weighted shift operators and analytic function theory" The 1980's saw the thriving of operator theoretic studies related to Bergman spaces The contributors in this period are numerous; their achievements were presented in Zhu's 1990 book "Operator Theory in Function Spaces" The research on Bergman spaces in the 1990 's resulted in several breakthroughs, both function theoretic and operator theoretic The most notable results in this period include Seip's geometric characterization of sequences of interpolation and sampling, Hedenmalm's discovery of the contractive zero divisors, the relationship between Bergman-inner functions and the biharmonic Green function found by vi Preface Duren, Khavinson, Shapiro, and Sundberg, and deep results concerning invariant subspaces by Aleman, Borichev, Hedenmalm, Richter, Shimorin, and Sundberg Our purpose is to present the latest developments, mostly achieved in the 1990's, in book form In particular, graduate students and new researchers in the field will have access to the theory from an almost self-contained and readable source Given that much of the theory developed in the book is fresh, the reader is advised that some of the material covered by the book has not yet assumed a final form The prerequisites for the book are elementary real, complex, and functional analysis We also assume the reader is somewhat familiar with the theory of Hardy spaces, as can be found in Duren's book "Theory of HP Spaces", Garnett's book "Bounded Analytic Functions", or Koosis' book "Introduction to if Spaces" Exercises are provided at the end of each chapter Some of these problems are elementary and can be used as homework assignments for graduate students But many of them are nontrivial and should be considered supplemental to the main text; in this case, we have tried to locate a reference for the reader We thank Alexandru Aleman, Alexander Borichev, Bernard Pinchuk, Kristian Seip, and Sergei Shimorin for their help during the preparation of the book We also thank Anders Dahlner for assistance with the computer generation of three pictures, and Sergei Treil for assistance with one January 2000 Haakan Hedenmalm Boris Korenblum Kehe Zhu Contents Preface v The Bergman Spaces Bergman Spaces 1.1 1.2 1.3 1.4 1.5 1.6 Some LP Estimates The Bloch Space Duality of Bergman Spaces Notes Exercises and Further Results The Berezin Transform 2.1 Algebraic Properties 2.2 Harmonic Functions 2.3 Carleson-Type Measures 2.4 BMO in the Bergman Metric 2.5 A Lipschitz Estimate 2.6 Notes 2.7 Exercises and Further Results A P -Inner Functions A~-Inner Functions An Extremal Problem The Biharmonic Green Function The Expansive Multiplier Property 3.1 3.2 3.3 3.4 13 17 22 23 28 28 32 38 42 46 49 49 52 52 55 59 66 Contents viii 3.5 3.6 3.7 3.8 3.9 Contractive Zero Divisors in A P An Inner-Outer Factorization Theorem for AP Approximation of Subinner Functions Notes Exercises and Further Results Zero Sets 4.1 Some Consequences of Jensen's Formula 4.2 Notions of Density The Growth Spaces A -Ci and A -00 • • 4.3 4.4 A -Ci Zero Sets, Necessary Conditions A -Ci Zero Sets, a Sufficient Condition 4.5 4.6 Zero Sets for Ag 4.7 The Bergman-Nevanlinna Class 4.8 Notes 4.9 Exercises and Further Results 71 78 86 94 95 98 98 104 110 112 119 128 131 133 134 Interpolation and Sampling Interpolation Sequences for A-Ci Sampling Sets for A-Ci Interpolation and Sampling in Ag Hyperbolic Lattices Notes Exercises and Further Results 136 136 152 155 165 171 172 Invariant Subspaces Invariant Subspaces of Higher Index 6.1 6.2 Inner Spaces in A~ A Beurling-Type Theorem 6.3 6.4 Notes Exercises and Further Results 6.5 176 176 180 181 186 187 5.1 5.2 5.3 5.4 5.5 5.6 7.1 7.2 7.3 7.4 7.5 7.6 Cyclicity Cyclic Vectors as Outer functions Cyclicity in A P Versus in A- oo Premeasures for Functions in A- oo Cyclicity in A- oo Notes Exercises and Further Results 190 190 191 193 208 214 214 Invertible Noncyclic Functions An Estimate for Harmonic Functions 8.1 8.2 The Building Blocks 8.3 The Basic Iteration Scheme 8.4 The Mushroom Forest 216 217 219 222 230 Contents 8.5 8.6 8.7 8.8 Finishing the Construction Two Applications Notes Exercises and Further Results ix 235 238 239 240 242 Logarithmically Subbarmonic Weights 9.1 Reproducing Kernels 9.2 Green Functions with Smooth Weights 9.3 Green Functions with General Weights 9.4 An Application 9.5 Notes 9.6 Exercises and Further Results 242 253 262 267 269 269 Rderences 274 Index 282 References 275 [13] S Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math J 53 (1986), 315-332 [14] S Axler, Bergman spaces and their operators, Surveys of Some Recent Results in Operator Theory, Volume 1, (J.B Conway and B.B Morrel, editors), Pitman Research Notes in Math 171, 1988, 1-50 [15] D Bekolle, C A Berger, L A Coburn, and K Zhu, BMO in the Bergman metric on bounded symmetric domains, J Funct Anal 93 (1990), 310-350 [16] H Bercovici, C Foia~, and C Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Mathematics, 56; published for the Conference Board of the Mathematical Sciences, Washington, D.C., by the American Mathematical Society, Providence, R.I., 1985 [17] F A Berezin, Covariant and contravariant symbols of operators, Matk USSR-Izv (1972),1117-1151 [18] F A Berezin, The relation between co- and contravariant symbols of operators on classical complex symmetric spaces, Soviet Matk Dokl 19 (1978), 786-789 [19] S Bergman, The kernel function and conformal mapping (second, revised edition), Mathematical Surveys and Monographs, vol 5, American Mathematical Society, Providence, RI, 1970 [20] B Berndtsson and J Ortega-Cerda, On interpolation and sampling in Hilbert spaces of analytic functions, J Reine Angew Math 464 (1995), 109-128 [21] A Beuding, On two problems concerning linear transformations in Hilbert space, Acta Math 81 (1949),239-255 [22] A Beurling, The Collected Works of Arne Beurling by L Carleson, P Molliavin, J Neuberger, and J Wermer, Volume 2, Harmonic Analysis, 341-365 Boston, Birkhauser, 1989 [23] C Bishop, Bounded functions in the little Bloch space, Pacific J Matk 142 (1990), 209-225 [24] G Bomash, A Blaschke-type product and random zero sets for the Bergman spaces, Ark Mat 30 (1992), 45-60 [25] A A Borichev, Estimates from below and cyclicity in Bergman-type spaces, Internat Math Res Notices 12 (1996), 603-611 [26] A Borichev, Invariant subspaces of given index in Banach spaces of analytic functions, J Reine Angew Matk 505 (1998), 23-44 [27] A Borichev and H Hedenmalm, Harmonic functions of maximal growth: invertibility and cyclicity in the Bergman spaces, J Amer Matk Soc 10 (1997), 761-796 [28] A Borichev, H Hedenmalm, and A Volberg, Large Bergman spaces: invertibility, cyclicity, and subspaces of arbitrary index, preprint, 2000 [29] P Bourdon, Similarity of parts to the whole for certain multiplication operators, Proc Amer Math Soc 99 (1987), 563-567 [30] L de Branges, A proof of the Bieberbach conjecture, Acta Math 154 (1985), 13 7-152 [31] L Brown and B Korenblum, Cyclic vectors in A -00 , Proc Amer Math Soc 102 (1988),137-138 [32] J Bruna and D Pascuas, Interpolation in A -00, J London Matk Soc (2) 40 (1989), 452-466 276 References [33] L Carieson, Sets of uniqueness for functions regular in the unit circle, Acta Math 87 (1952),325-345 [34] R R Coifman, R Rochberg, and G Weiss, Factorization theorems for Hardy spaces in several variables, Ann of Math (2) 103 (1976), 611-635 [35] D S Cyphert and J A Kelingos, The decomposition of bounded K-variation into differences of K-decreasing functions, Studia Math 81 (1985), 185-195 [36] A E Djrbashian and F A Shamoian, Topics in the theory of Ag spaces, TeubnerTexte zur Mathematik [Teubner Texts in Mathematics], 105, BSB B G Teubner Verlagsgesellschaft, Leipzig, 1988 [37] P L Duren, Theory of H P spaces, Pure and Applied Mathematics, Vol 38, Academic Press, New York-London, 1970 [38] P Duren, D Khavinson, H S Shapiro, and C Sundberg, Contractive zero-divisors in Bergman spaces, Pacific J Math 157 (1993), 37-56 [39] P Duren, D Khavinson, H S Shapiro, and C Sundberg, Invariant subspaces and the biharmonic equation, Michigan Math J 41 (1994),247-259 [40] P Duren, D Khavinson, and H S Shapiro, Extremal functions in invariant subspaces of Bergman spaces, Illinois J Math 40 (1996), 202-210 [41] P L Duren, B W Romberg, and A L Shields, Linearfunctionals on HP spaces with o < p < 1, J Reine Angew Math 238 (1969), 32-60 [42] M EngliS, Toeplitz operators on Bergman-type spaces, Kandidatska disertacni prace (PhD thesis), MU CSAV, Praha, 1991 [43] M Englis, Functions invariant under the Berezin transform, Funct Anal 121 (1994), 233-254 [44] M EngliS, A Loewner-type lemma for weighted biharmonic operators, Pacific J Math 179 (1997), 343-353 [45] M Englis, Weighted biharmonic Green functions for rational weights, Glasg Math J 41 (1999),239-269 [46] C Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent Math 26 (1974),1-65 [47] F Forelli and W Rudin, Projections on spaces of holomorphic functions on balls, Indiana Univ Math J 24 (1974), 593-602 [48] D Gale, H W Kuhn, and A W Tucker, Linear programming and the theory of games, Activity Analysis of Production and Allocation, Cowles Commission Monograph No 13 John Wiley & Sons, Inc., New York, N Y.,; Chapman & Hall, Ltd., London, 1951; pp.317-329 [49] J Garnett, Bounded analyticjunctions, Pure and Applied Mathematics, 96, Academic Press, Inc., New York-London, 1981 [50] J Hadamard, (Euvres de Jacques Hadamard, Vols 1-4, Editions du Centre National de la Recherche Scientifique, Paris, 1968 [51] P Halmos, Shifts on Hilbert spaces, Reine Angew Math 208 (1961),102-112 [52] J Hansbo, Reproducing kernels and contractive divisors in Bergman spaces, Zap Nauchn Sem S.-Peterburg Otdel Mat Inst Steklov (POM!) 232 (1996), Issled po Linein Oper i Teor Funktsii, 24,174-198 References 277 [53] G H Hardy and J E Littlewood, Some properties offractional integrals, Math Z 34 (1932),403-439 [54] w W Hastings, A Carleson measure theorem for Bergman spaces, Proc Amer Math Soc 52 (1975), 237-24l [55] V P Havin, N K Nikolski (editors), Linear and complex analysis, problem book 3, Part II, Lecture Notes in Mathematics 1574, Springer-Verlag, Berlin, 1994 [56] W K Hayman, Subharmonicjunctions, Vol 2, London Mathematical Society Monographs, 20, Academic Press, Inc [Harcourt Brace Jovanovich, Publishers], London, 1989 [57] W K Hayman, On a conjecture of Koren blum Analysis (Munich) 19 (1999), 195-205 [58] w K Hayman and P B Kennedy, Subharmonicjunctions, Vol 1, London Mathematical Society Monographs, 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976 [59] H Hedenmalm, A factorization theorem for square area-integrable analytic functions, J Reine Angew Math 422 (1991), 45-68 [60] H Hedenmalm, A factoring theorem for a weighted Bergman space, St Petersburg Math J (1993),163-174 [61] H Hedenmalm, An invariant subspace of the Bergman space having the codimension two property, J Reine Angew Math 443 (1993), 1-9 [62] H Hedenmalm, Spectral properties of invariant subspaces in the Bergman space, J Funct Anal 116 (1993), 441-448 [63] H Hedenmalm, A factoring theorem for the Bergman space, Bull London Math Soc 26 (1994), 113-126 [64] H Hedenmalm, A computation of the Green function for the weighted biharmonic operators ~lzl-2a~, with a > -1, Duke Math J 75 (1994), 51-78 [65] H Hedenmalm, Open problems in the function theory of the Bergman space, Festschrift in honour of Lennart Carleson and Yngve Domar (Uppsala, 1993), 153169, Acta Univ Upsaliensis Skr Uppsala Univ C Organ Hist., 58, Uppsala Univ., Uppsala, 1995 [66] H Hedenmalm, Boundary value problems for weighted biharmonic operators, St Petersburg Math J (1997), 661-674 [67] H Hedenmalm, Maximal invariant subspaces in the Bergman space Ark Mat 36 (1998),97-1Ol [68] H Hedenmalm, An off-diagonal estimate of Bergman kernels, J Math Pures Appl., to appear [69] H Hedenmalm, S Jakobsson, and S Shimorin, A biharmonic maximum principle for hyperbolic surfaces, preprint, 2000 [70] H Hedenmalm, B Korenblum, and K Zhu, Beurling type invariant subspaces of Bergman spaces, J London Math Soc (2) 53 (1996), 601-614 [71] H Hedenmalm, S Richter, and K Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J Reine Angew Math 477 (1996), 13-30 [72] H Hedenmalm and S Shimorin, Hele-Shaw flow on hyperbolic surfaces, preprint, 2000 278 References [73] H Hedenmalm and K Zhu, On the failure of optimal factorization for certain weighted Bergman spaces, Complex Variables Theory Appl 19 (1992), 165-176 [74] A Heilper, The zeros offunctions in Nevanlinna's area class,lsraelJ Math 34 (1979), 1-11 [75] A Hinkkanen, A remark on a maximum principle in Bergman space, preprint, 1999 [76] C Horowitz, Zeros of functions in the Bergman spaces, Duke Math 41 (1974), 693-710 [77] C Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math 1.44 (1977), 201-213 [78] c Horowitz, B Korenblum, and B Pinchuk, Sampling sequences for A -00, Michigan Math 44 (1997), 389-398 [79] E L Ince, Ordinary differential equations, Dover Publications, New York, 1944 [80] J Janas, A note on invariant subspaces under multiplication by z in Bergman space, Proc Roy Irish Acad 83A (1983),157-164 [81] D Khavinson and H S Shapiro, Invariant subspaces in Bergman spaces and Hedenmalm's boundary value problem, Ark Mat 32 (1994), 309-321 [82] P Koosis, Introduction to Hp spaces, Second edition (with two appendices by V P Havin), Cambridge Tracts in Mathematics, 115 Cambridge University Press, Cambridge, 1998 [83] B Korenblum, An extension of the Nevanlinna theory, Acta Math 135 (1975), 187219 [84] B Korenblum, A Beurling-type theorem, Acta Math 138 (1977), 265-293 [85] B Korenblum, Cyclic elements in some spaces of analytic functions, Bull Amer Math Soc (1981), 317-318 [86] B Korenblum, Transformation of zero sets by contractive operators in the Bergman space, Bull Sci Math 114 (1990),385-394 [87] B Korenblum, A maximum principle for the Bergman space, Publ Mat 35 (1991), 479-486 [88] B Korenblum, Outer functions and cyclic elements in Bergman spaces, Funct Anal 115(1993),104-118 [89] B Korenblum and K Zhu, An application ofTauberian theorems to Toeplitz operators, Operator Theory 33 (1995), 353-361 [90] E LeBlanc, A probabilistic zero set condition for Bergman spaces, Michigan Math 37 (1990), 427-438 [91] K Lowner, Untersuchungen tiber schlichte konforme Abbildungen des Einheitskreises, Math Ann 89 (1923), 103-121 [92] D H Luecking, A technique for characterizing Carleson measures on Bergman spaces, Proc Amer Math Soc 87 (1983), 656-660 [93] D H Luecking, Zero sequences for Bergman spaces, Complex Variables Theory Appl 30 (1996), 345-362 [94] T H MacGregor and M Stessin, Weighted reproducing kernels in Bergman spaces, Michigan Math 41 (1994),523-533 [95] T H MacGregor and K Zhu, Coefficient multipliers between Bergman and Hardy spaces, Mathematika 42 (1995), 413-426 References 279 [96] X Massaneda, Density Conditions for Interpolation in A -00, preprint, 1999 [97] Z Nehari, Conformal mapping (Reprinting of the 1952 McGraw-Hill edition), Dover Publications, Inc., New York, 1975 [98] M Renardy, R.C Rogers, An introduction to partial differential equations, Texts in Applied Mathematics, 13, Springer-Verlag, New York, 1993 [99] S Richardson, Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J Fluid Mech 56 (1972), 609-618 [100] S Richter, Invariant subspaces in Banach spaces of analytic functions, Trans Amer Math Soc 304 (1987), 585-616 [101] S Richter, A representation theorem for cyclic analytic two-isometries, Trans Amer Math Soc 328 (1991), 325-349 [102] J Roberts, Cyclic inner functions in Bergman spaces and weak outer functions in H P, < P < I, Illinois J Math 29 (1985), 25-38 [103] R Rochberg, Interpolation by functions in Bergman spaces, Michigan Math J 29 (1982),229-236 [104] L Rubel and R Timoney, An extremal property of the Bloch space, Proc Amer Mafh Soc 75 (1979), 45-49 [105] W Rudin, Function theory in the unit ball o/en , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 241 SpringerVerlag, New York-Berlin, 1980 [106] W Rudin, Functional analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991 [107] S Saitoh, Theory o/reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, 189, Longman Scientific & Technical, Harlow, 1988 [108] M Sakai, Regularity of a boundary having a Schwarz function, Acta Math 166 (1991),263-297 [109] A P Schuster and K Seip, A Carleson-type condition for interpolation in Bergman spaces, J Reine Angew Math 497 (1998), 223-233 [110] A P Schuster and K Seip, Weak conditions for interpolation in holomorphic spaces, preprint, 1999 [III] K Seip, Regular sets of sampling and interpolation for weighted Bergman spaces, Proc Amer Math Soc 117 (1993), 213-220 [112] K Seip, Beurling type density theorems in the unit disk, Invent Math 113 (1993), 21-39 [113] K Seip, On Korenblum's density condition for zero sequences of A -a, J Anal Math 67 (1995), 307-322 [114] H S Shapiro, Some remarks on weighted polynomial approximation of hoi om orphic functions, Math USSR-Sb (1967), 285-294 [115] J H Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on Hardy and Bergman spaces, Duke Math J 43 (1976),187-202 [116] J H Shapiro, Zeros of random functions in Bergman spaces, Ann Inst Fourier (Grenoble) 29 (1979), no 4, vii, 159-171 [117] J H Shapiro, Cyclic innerfunctions in Bergman spaces, manuscript, 1980 280 References [118] A L Shields, Weighted shift operators and analytic function theory Topics in operator theory, Math Surveys, No 13, Amer Math Soc., Providence, R.I., 1974, pp 49-128 [119] A L Shields, Cyclic vectors in Banach spaces of analytic functions, Operators and function theory (S.C Power, editor; Lancaster, 1984) NATO Adv Sci Inst Ser C: Math Phys Sci., 153, D Reidel, Dordrecht-Boston, Mass., 1985, pp 315-349 [120] S M Shimorin, Factorization of analytic functions in weighted Bergman spaces, St Petersburg Math (1994),1005-1022 [121] S M Shimorin, On a family of conformally invariant operators, St Petersburg Math (1996), 287-306 [122] S M Shimorin, The Green function for the weighted biharmonic operator ~(1 IzI2)-a ~ and the factorization of analytic functions (Russian), Zap Nauchn Sem S.-Peterburg Otdel Mat Inst Steklov (POM!) 222 (1995), Issled po Linein Oper i Teor Funktsii, 23, 203-221 [123] S M Shimorin, Single point extremal functions in weighted Bergman spaces, Nonlinear boundary-value problems and some questions offunction theory, Math Sci 80 (1996), 2349-2356 [124] S M Shimorin, The Green functions for weighted biharmonic operators of the form ~ w -1 ~ in the unit disk, Some questions ofmathematical physics and function theory Math Sci (New York) 92 (1998), 4404-4411 [125] S M Shimorin, Approximate spectral synthesis in Bergman spaces, Duke Math 101 (2000), 1-40 [126] S M Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries; submitted for publication, 1999 [127] J Stoer and C Witzgall, Convexity and optimization in finite dimensions, I, Grundlehren der mathematischen Wissenschaften, Band 163 Springer-Verlag, New York-Berlin, 1970 [128] c Sundberg, Analytic continuability of Bergman inner functions, Michigan Math 1.44 (1997), 399-407 [129] J E Thomson, Approximation in the mean by polynomials, Ann of Math (2) 133 (1991),477-507 [130] J E Thomson, Bounded point evaluations and polynomial approximation, Proc Amer Math Soc 123 (1995),1757-1761 [131] D Vukotic, A sharp estimate for Ag functions in (1993), 753~756 en, Proc Amer Math Soc 117 [132] K Zhu, VMO, ESV, and Toeplitz operators on the Bergman space, Trans Amer Math 302 (1987), 617-646 [133] K Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, Operator Theory 20 (1988), 329-357 [134] K Zhu, Multipliers of BMO in the Bergman metric with applications to Toeplitz operators, Funct Anal 87 (1989), 31-50 [135] K Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990 [136] K Zhu, Bergman and Hardy spaces with small exponents, Pacific Math 162 (1994),189-199 [137] K Zhu, Distances and Banach spaces of hoi omorphic functions on complex domains, J London Math Soc 49 (1994), 163-182 [138] K Zhu, Interpolating sequences for the Bergman space Michigan Math J 41 (1994), 73-86 [139] K Zhu, Evaluation operators on the Bergman space, Math Proc Cambridge Phi/os Soc 117 (1995), 513-523 [140] K Zhu, Interpolating and recapturing in reproducing Hilbert spaces, Bull Hong Kong Math Soc (1997),21-33 [141] K Zhu, Restriction of the Bergman shift to an invariant subspace, Quart J Math Oxford 48 (1997), 519-532 [142] K Zhu, A class of operators associated with reproducing kernels, J Operator Theory 38 (1997), 19-24 [143] K Zhu, Maximal inner spaces and Hankel operators on the Bergman space, Integral Equations Operator Theory 31 (1998), 371-387 [144] K Zhu, A sharp estimate for extremal functions, Proc Amer Math Soc., in print Index A2 -subinner function, 52 AP-inner function, 52 Ag+, limit of Bergman spaces, 130 Ag-, limit of Bergman spaces, 130 B2(w), large weighted Bergman space, 245 B p, the Besov space, 23 C k (X),60 C a (r),137 D(z, r), hyperbolic disk, 38 D+(A), upper asymptotic K-density, o 105 D;(r), upper Seip density, 147 D;t(r), uniform separating upper asymptotic K-density of f, 140 D-(A), lower asymptotic K-density, 105 D-; (f), lower Seip density, 153 Da, "fractional differentiation", 18 Da, "fractional integration", 19 G(z, w), Green function for the Laplacian, 60 G[f], Green potential, 60 G A, zero divisor for A, 57 G /, the extremal function for I, 57 H P, the Hardy space, 54 BeAz, w), the harmonic compensator, 266 Hw.D, harmonic compensator, 254 lA, zero-based invariant subspace, 55 If, invariant subspace generated by /,56 K(z, w), the Bergman kernel, Ka(z, w), the Bergman kernel for standard weights, Kw(z, w), weighted Bergman kernel, 245 N(r), counting function, 100 P(z, w), Poisson kernel, 29 P(z, w), the Poisson kernel, 114,217 P*, the sweep operator, 87 Qw(z, w), weighted harmonic Bergman kernel, 252 13, the Bloch space, 13 130, the little Bloch space, 13 C, the complex plane, lDJ, the open unit disk, 6., the Laplacian, 32 fez, w), biharmonic Green function, 62 Index r[n, biharmonic Green potential, 62 r (J), the weighted biharmonic Green function, 242 r a(Z), approximate Stolz angle, 111 JR, the real line, T, the unit circle, A-oo, the big growth space, 110 A -a, standard growth space, 110 A+a, limit of growth spaces, 112 A=a, limit of growth spaces, 112 Aoa, small growth space, 110, 137 A P, the Bergman spaces, A P -outer function, 190, 191 A~, the standard weighted Bergman space, A~-inner function, 52 A~-subinner function, 87 A~, the Bergman-Nevanlinna class, 98, 131 A2(w), weighted Bergman space, 245 A2(w)-inner function, 270 B, the Berezin transform, 29 Ba, the weighted Berezin transform, 29 {3, the hyperbolic metric, 16 V, the Dirichlet space, 25 C(]),13 Co(IDl), 13 oCT), diagonal, 253 HP2(w),251 MO(f), "mean oscillation", 43 MO r (f), mean oscillation, 42 H(IDl), the space of all holomorphic functions, 18 Hoo,2 KB+,194 K·-absolutely continuous, 201 K·-area, 148 K·-bound,194 K·-bounded above, 194 K·-density, 105 K -smooth, 201 K-variation, 194 A, invariant Laplacian, 32 V, the gradient, 51 283 w, harmonic measure, 218 w, weight, 242, 244 w-mean value disk, 255 ~(A, E), partial Blaschke sum on E, 105 A(A, E), partial logarithmic Blaschke sum on E, 105 P, Bergman projection, Pa, weighted Bergman projection, p, the pseudohyperbolic metric, 16 ~,2 Stolz star, 105 Stolz angle with aperture ct, 106 Sz, Stolz angle, 104 rpz, the Mobius involution, W"r, harmonic measure, 255 KCF), Beurling-Carleson characteristic, 104 dA, the normalized area element, dAa, the weighted area element, dAE, push-out measure, 114 dc, Euclidean metric on the plane, 105 d1f, circle metric, 104 ds, normalized arc length measure, 59, 104 nCr), counting function, 100 n I , common multiplicity of zero at the origin, 56 BMOa,45 BMO r ,42 VMOa,46 VMO r ,46 "inner-outer" factorization, 78, 186, 239 SF, Sz,a, absolutely continuous measure, 201 admissible, 122 analytic projection, 12 aperture, 104, 106 arc length, 104 arc length normalized, 104 arithmetic-geometric mean inequality, 101 284 Index asymptotic K-density lower, 105 upper, 105 atomic singularity, 59 balayage-type estimate, 112 Berezin transform, 28 Bergman kernel, Bergman metric, 16 Bergman projection, Bergman space, Bergman-Nevanlinna class, 98, 131 Besov space, 24, 50 Beurling's theorem, 176 Beurling-Carleson characteristic, 104 Beurling-Carleson set, 104,200 Beurling-type theorem, 181 bi-harmonic Green function, 62 bi-harmonic Green potential, 62 biharmonic Green function, 59, 242 Blaschke product, 26 Bloch space, 13 BMO,42 BMO in the Bergman metric, 42 BMOA,13 bounded K-variation, 194 building blocks, 219 Caratheodory-Schur theorem, 87 Carleson measure, 38 Carleson square, 106, 240 Cayley transform, 113, 165, 167 complementary arcs, 104 concave operator, 182 concave sequence, 183 concavity-type property of the biharmonic Green function, 273 conjugate exponent, 18, 160 contractive divisibility property, 59 contractive mUltiplier, 54 contractive zero divisors, 71 covering, 125,204 cyclic function, 190 cyclic invariant subspace, 56 cyclic vector, 78,85, 190 density, 104 Dirichlet space, 25 division property, 189 domination, 190, 191 duality theorem of Linear Programming, 124,204 edge point in polyhedron, 124 eigenfunctions, 36 eigenvalue, 37 elementary factors, 131 entropy, 104 exact type, 111 expansive multiplier property, 59, 66, 70 extraneous zero, 72 extremal function, 57 extremal problem, 26, 55, 56, 195 extremal problem for I, 56 factorization, 78 Fejer kernel, 87 fractional differentiation, 18 fractional integration, 18 functions of bounded variation, 195 gap sequence, 15 Garsia's lemma, 42 generation, 233 geometric progression, 107 Green function, 59 Green potential, 60 Green's formula, 59 growth space, 110 Hadamard's variational formula, 255, 257 Hankel operator, 49 Hardy space, 12,54,78 harmonic compensator, 254, 266 harmonic conjugate, 26 harmonic majorant, 55 Index harmonic measure, 218, 219, 225, 232 Helly selection theorem, 195 Herglotz measure, 201 Herg10tz transform, 259 homogeneous decomposition, 178 Hurwitz's theorem, 77 hyperbolic center, 42 hyperbolic disk, 38 hyperbolic exponential type, 111 hyperbolic metric, 16 hyperbolic radius, 42 index, 176 index n, 176 index of an invariant subspace, 176 inner function, 52, 78 inner function for Bergman spaces, 53 inner space, 180 inner-outer factorization, 78 interpolating sequence, 50, 136 interpolation problem, 137 invariant Laplacian, 32 invariant subspace, 55 invariant subspace problem, 187 iteration scheme, 222 Jensen's formula, 98 Jensen-type inequality, 116 Krebe function, 260 lacunary series, 15 Laplace equation, 62 Laplace-Beltrami operator, 33 Laplacian, 32 Laplacian invariant, 32 Linear Programming, 119 Linear Programming duality theorem, 124, 204 little Bloch space, 13 localization trick, 230 logarithmic entropy, 194 logarithmically sUbharmonic, 94, 242, 244 285 lower asymptotic K-density, 105 lower Seip density, 153 lunula,227 Mobius group, Mobius map, Mobius transformation, 16 maximal inner space, 187 mean oscillation, 42 min-max equation, 123 minimal type, 111 mushroom, 231 mushroom forest, 230 mushroom hat, 232 mushroom stem, 231 nonoverlapping arcs, 211 normalized arc length, 59, 104 oblique projection, 119 optimization problem, 123 outer function, 190 perturbation, 139 Poincare metric, 16 point-evaluation, Poisson extension, 114 Poisson formula, 29 Poisson kernel, 29, 114 Poisson solver, 273 Poisson transform, 28, 29, 114,273 positive function, measure, premeasure, 190, 193 pseudohyperbolic metric, 16 push-out measure, 114 quasi-Banach space, quasi-similar operators, 188 regular sequence, 177 reproducing for the origin, 242 reproducing kernel, 57, 242 residue theorem, 76 286 Index restriction operator, 155 reverse triangle inequality, 177 sampling sequence, 136 Schur's test, 9, 11 Seip density lower, 153 Seip density upper, 147 separated sequence, 39,50, 138 sequence of interpolation, 136 sequence of sampling, 136 sesquiholomorphic, 252 similar operators, 188 simple covering, 125,204 singly generated invariant subspace, 56 singular inner function, 59 singular measure, 201 spectral mapping theorem, 37 Stieltjes integral, 195 Stolz angle, 104 Stolz star, 105 strictly positive, subinner function, 86 suppressive weight, 273 sweep of a function, 88, 264 Toeplitz operator, 49 truncation, 202 type, hyperbolic exponential, III uniform separating upper asymptotic K-density, 140 uniform upper asymptotic K-*-density, 146 uniformly discrete, 39 upper asymptotic K-density, 105 upper Seip density, 147 variational argument, 56 variational formula, 255 VMO,42 weakly cyclic, 201 Weierstrass factorization, 131 weighted Bergman kernel, weighted Bergman projection, weighted Bergman space, weighted biharmonic Green function, 242 weighted Hele-Shaw flows, 255 Wirtinger derivatives, zero divisor, 57 zero sequence, 98 zero set, 98 zero-based invariant subspace, 56 (continued from page ii) 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 18 79 80 811 82 83 84 85 86 87 88 89 90 91 92 KARGA1'oLOv/MERLZJAKOV Fundamentals of the Theory of Groups BOLLOBAS Graph Theory EDWARDS Fourier Series Vol I 2nd ed WELLS Differential Analysis on Complex Manifolds 2nd ed WATERHOUSE Introduction to Affine Group Schemes SERRE Local Fields WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields II MASSEY Singular Homology Theory F ARKAslKRA Riemann Surfaces 2nd ed STILLWELL Classical Topology and Combinatorial Group Theory 2nd ed HUNGERFORD Algebra DAVENPORT Multiplicative Number Theory 2nd ed HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras IITAKA Algebraic Geometry HECKE Lectures on the Theory of Algebraic Numbers BURRIS/SANKAPPANAVAR A Course in Universal Algebra WALTERS An Introduction to Ergodic Theory ROBINSON A Course in the Theory of Groups 2nd ed FORSTER Lectures on Riemann Surfaces BOTT/TU Differential Forms in Algebraic Topology WASHINGTON Introduction to Cyclotomic Fields 2nd ed IRELANDIROSEN A Classical Introduction to Modern Number Theory 2nd ed EDWARDS Fourier Series Vol II 2nd ed VAN LINT Introduction to Coding Theory 2nd ed BROWN Cohomology of Groups PIERCE Associative Algebras LANG Introduction to Algebraic and Abelian Functions 2nd ed BR0NDSTED An Introduction to Convex Polytopes BEARDON On the Geometry of Discrete Groups DIES TEL Sequences and Series in Banach Spaces 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 III 112 113 114 115 116 117 118 119 DUBROVINlFoMENKOlNovIKOV Modern Geometry-Methods and Applications Part I 2nd ed WARNER Foundations of Differentiable Manifolds and Lie Groups SHIRYAEV Probability 2nd ed CONWAY A Course in Functional Analysis 2nd ed KOBLITZ Introduction to Elliptic Curves and Modular Forms 2nd ed BROCKERIToM DIECK Representations of Compact Lie Groups GROVEIBENSON Finite Reflection Groups 2nd ed BERG/CHRlSTENSENiRESSEL Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions EDWARDS Galois Theory V ARADARAJAN Lie Groups, Lie Algebras and Their Representations LANG Complex Analysis 3rd ed DUBROVINlFoMENKOINOVIKOV Modern Geometry-Methods and Applications Part II LANG SLiR) SILVERMAN The Arithmetic of Elliptic Curves OLVER Applications of Lie Groups to Differential Equations 2nd ed RANGE Holomorphic Functions and Integral Representations in Several Complex Variables LEHTO Univalent Functions and Teichmiiller Spaces LANG Algebraic Number Theory HUSEMOLLER Elliptic Curves LANG Elliptic Functions KAKATZAS/SHREVE Brownian Motion and Stochastic Calculus 2nd ed KOBLITZ A Course in Number Theory and Cryptography 2nd ed BERGERIGOSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces KELLEy/SRINIVASAN Measure and Integral Vol.L SERRE Algebraic Groups and Class Fields PEDERSEN Analysis Now ROTMAN An Introduction to Algebraic Topology 120 ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields I and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBlNGHAUS/HERMES et al Numbers Readings in Mathematics 124 DUSROVlNlFoMENKOINOVIKOV Modern Geometry-Methods and Applications Part III 125 BERENSTElN/GAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 2nd ed 127 MASSEY A Basic Course in Algebraic Topology 128 RAUCH Partial Differential Equations 129 FULTON/HARRIs Representation Theory: A First Course Readings in Mathematics 130 DODSONIPOSTON Tensor Geometry 131 LAM A First Course in Nonconunutative Rings 132 BEARDON Iteration of Rational Functions 133 HARRIs Algebraic Geometry: A First Course 134 ROMAN Coding and Infonnation Theory 135 ROMAN Advanced Linear Algebra 136 ADKlNsIWElNTRAUB Algebra: An Approach via Module Theory 137 AXLERIBOURDON/RAMEy Harmonic Function Theory 138 COHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKERIWEISPFENNlNGIKREDEL Grabner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DOOB Measure Theory 144 DENNIS/F ARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K-Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds ISO EISENBUD Conunutative Algebra with a View Toward Algebraic Geometry lSI SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A First Course 154 BROWNIPEARCY An Introduction to Analysis 155 KASSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIAVIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEINIERDELYI Polynomials and Polynomial Inequalities 162 ALPERINIBELL Groups and Representations 163 DIXONIMORTIMER Pennutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Cartan's Generalization of Klein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed 171 PETERSEN Riemannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 DIESTEL Graph Theory 2nd ed 174 BRIDGES Foundations of Real and Abstract Analysis 175 LICKORISH An Introduction to Knot Theory 176 LEE Riemannian Manifolds 177 NEWMAN Analytic Number Theory 178 CLARKE/LEDYAEVISTERNIWOLENSKl Nonsmooth Analysis and Control Theory 179 DOUGLAS Banach Algebra Techniques in Operator Theory 2nd ed 180 SRIVASTAVA A Course on Borel Sets 181 KREss Numerical Analysis 182 WALTER Ordinary Differential Equations 183 MEGGINSON An Introduction to Banach Space Theory 184 BOLLOBAS Modem Graph Theory 185 COxILITTLE/O'SHEA Using Algebraic Geometry 186 RAMAKRlSHNAN/V ALENZA Fourier Analysis on Number Fields 187 HARRIS/MORRISON Moduli of Curves 188 GOLDBLATT Lectures on the Hyperreals: An Introduction to Nonstandard Analysis 189 LAM Lectures on Modules and Rings 190 ESMONDE/MuRTY Problems in Algebraic Number Theory 191 LANG Fundamentals of Differential Geometry 192 HIRSCHILACOMBE Elements of Functional Analysis 193 COHEN Advanced Topics in Computational Number Theory 194 ENGELiNAGEL One-Parameter Semigroups for Linear Evolution Equations 195 NATHANSON Elementary Methods in Number Theory 196 OSBORNE Basic Homological Algebra 197 EISENBUD/ HARRIS The Geometry of Schemes 198 ROBERT A Course in p-adic Analysis 199 HEDENMALMIKORENBLUMIZHU Theory of Bergman Spaces 200 BAO/CHERN/SHEN An Introduction to Riemann-Finsler Geometry 201 HINDRY/SILVERMAN Diophantine Geometry: An Introduction 202 LEE Introduction to Topological Manifolds ... WHITEHEAD Elements of Homotopy (continued after index) Haakan Hedenmalm Boris Korenblum Kehe Zhu Theory of Bergman Spaces With Illustrations Springer Haakan Hedenmalm Department of Mathematics Lund... (2000): 47-01, 47A15, 32A30 Library of Congress Cataloging-in-Publication Data Hedenmalm, Haakan Theory of Bergman spaces I Haakan Hedenmalrn, Boris Korenblurn, Kehe Zhu p cm - (Graduate texts in rnathernatics... the subspace of B consisting of functions I with lim (1 Izl-+l- -ld)I/'(z)1 = o The Bloch space plays the same role in the theory of Bergman space as the space BMOA does in the theory of Hardy spaces