Spaces of holomorphic functions have been a prominent theme in analysis since early in the twentieth century. Of interest to complex analysts, functional analysts, operator theorists, and systems theorists, their study is now flourishing. This volume, an outgrowth of a 1995 program at the Mathematical Sciences Research Institute, contains expository articles by program participants describing the present state of the art. Here researchers and graduate students will encounter Hardy spaces, Bergman spaces, Dirichlet spaces, Hankel and Toeplitz operators, and a sampling of the role these objects play in modern analysis.
Preface; 1. Holomorphic spaces: a brief and selective survey Donald Sarason; 2. Recent progress in the function theory of the Bergman space Håkan Hedenmalm; 3. Harmonic Bergman spaces Karel Stroethoff; 4. An excursion into the theory of Hankel operators Vladimir V. Peller; 5. Hankel-type operators, Bourgain algebras and uniform algebras Pamela Gorkin; 6. Tight uniform algebras Scott Saccone; 7. Higher-order Hankel forms and commutators Richard Rochberg; 8. Function theory and operator theory on the Dirichlet space Zhijian Wu; 9. Some open problems in the theory of subnormal operators John B. Conway and Liming Yang; 10. Elements of spectral theory in terms of the free function model part I: basic constructions Nikolai Nikolski and Vasily Vasyunin; 11. Liftings of kernels shift-invariant in scattering systems Cora Sadosky; 12. Some function-theoretic issues in feedback stabilisation Nicholas Young; 13. The abstract interpolation problem Alexander Kheifets; 14. A basic interpolation problem Harry Dym; 15. Reproducing kernel Pontryagin spaces Daniel Alpay, Aad Dijksma, James Rovnyak and Hendrik S. V. de Snoo; 16. Commuting operators and function theory on a Riemann surface Victor Vinnikov.