Theory of Bergman Spaces (Graduate Texts in Mathematics Book 199)

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. Fifteen years ago, most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely, yet today the situation has completely changed. For several years, research interest and activity have expanded in this area and there are now rich theories describing the Bergman spaces and their operators. This book is a timely treatment of the theory, written by three of the major players in the field. Front Matter....Pages i-ix The Bergman Spaces....Pages 1-27 The Berezin Transform....Pages 28-51 A p -Inner Functions....Pages 52-97 Zero Sets....Pages 98-135 Interpolation and Sampling....Pages 136-175 Invariant Subspaces....Pages 176-189 Cyclicity....Pages 190-215 Invertible Noncyclic Functions....Pages 216-241 Logarithmically Subharmonic Weights....Pages 242-273 Back Matter....Pages 274-289 1. The Bergman Spaces -- 2. The Berezin Transform -- 3. A[superscript P]-inner Functions -- 4. Zero Sets -- 5. Interpolation And Sampling -- 6. Invariant Subspaces -- 7. Cyclicity -- 8. Invertible Noncyclic Functions -- 9. Logarithmically Subharmonic Weights. Hakan Hedenmalm, Boris Korenblum, Kehe Zhu. Includes Bibliographical References And Index. The Bergman Spaces The Berezin Transform A^p-Inner Functions Zero Sets for Bergman Spaces Invariant Subspaces Interpolation and Sampling Cyclic Vectors. In this chapter we introduce the Bergman spaces and concentrate on the general aspects of these spaces.