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Preliminary Text. Do not use. 15 years ago the function theory and operator theory connected with the Hardy spaces was well understood (zeros; factorization; interpolation; invariant subspaces; Toeplitz and Hankel operators, etc.). None of the techniques that led to all the information about Hardy spaces worked on their close relatives the Bergman spaces. Most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely. Now the situation has completely changed. Today there are rich theories describing the Bergman spaces and their operators. Research interest and research activity in the area has been high for several years. A book is badly needed on Bergman spaces and the three authors are the right people to write it.
Content Level »Graduate
Keywords »Function Theory - Nevanlinna theory - Operator theory - algebra - bounded mean oscillation - extrema - logarithm
1 The Bergman Spaces.- 1.1 Bergman Spaces.- 1.2 Some Lp Estimates.- 1.3 The Bloch Space.- 1.4 Duality of Bergman Spaces.- 1.5 Notes.- 1.6 Exercises and Further Results.- 2 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.- 3 Ap -Inner Functions.- 3.1 Ap? -Inner Functions.- 3.2 An Extremal Problem.- 3.3 The Biharmonic Green function.- 3.4 The Expansive Multiplier Property.- 3.5 Contractive Zero Divisors in Ap.- 3.6 An Inner-Outer Factorization Theorem for Ap.- 3.7 Approximation of Subinner Functions.- 3.8 Notes.- 3.9 Exercises and Further Results.- 4 Zero Sets.- 4.1 Some Consequences of Jensen’s Formula.- 4.2 Notions of Density.- 4.3 The Growth Spaces A-? and A-?.- 4.4 A-? Zero Sets, Necessary Conditions.- 4.5 A-? Zero Sets, a Sufficient Condition.- 4.6 Zero Sets for AP?.- 4.7 The Bergman-Nevanlinna Class.- 4.8 Notes.- 4.9 Exercises and Further Results.- 5 Interpolation and Sampling.- 5.1 Interpolation Sequences for AT-?.- 5.2 Sampling Sets for A-?.- 5.3 Interpolation and Sampling in Ap?.- 5.4 Hyperbolic Lattices.- 5.5 Notes.- 5.6 Exercises and Further Results.- 6 Invariant Subspaces.- 6.1 Invariant Subspaces of Higher Index.- 6.2 Inner Spaces in A2?.- 6.3 A Beurling-Type Theorem.- 6.4 Notes.- 6.5 Exercises and Further Results.- 7 Cyclicity.- 7.1 Cyclic Vectors as Outer functions.- 7.2 Cyclicity in Ap Versus in A-?.- 7.3 Premeasures for Functions in A-?.- 7.4 Cyclicity in A-?.- 7.5 Notes.- 7.6 Exercises and Further Results.- 8 Invertible Noncyclic Functions.- 8.1 An Estimate for Harmonic Functions.- 8.2 The Building Blocks.- 8.3 The Basic Iteration Scheme.- 8.4 The Mushroom Forest.- 8.5 Finishing the Construction.- 8.6 Two Applications.- 8.7 Notes.- 8.8 Exercises and Further Results.- 9 Logarithmically Subharmonic 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.- References.