#33: Holomorphic Spaces
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MSRI Publications -- Volume 33
Holomorphic Spaces
Edited by Sheldon Axler, John McCarthy, and Donald Sarason
The term "Holomorphic Spaces" is short
for "Spaces of Holomorphic Functions". It refers not so much to a
branch
of mathematics as to a common thread running through much of modern
analysis---through functional analysis, operator theory, harmonic analysis,
and, of course, complex analysis.
In the fall of 1995 the Mathematical Sciences Research Institute in
Berkeley
sponsored the program Holomorphic Spaces. Over forty participants
came for
periods of two weeks to a full semester; an additional forty or so
attended
a week-long workshop in October. Spaces of holomorphic functions
arise in
many contexts. The MSRI program focused predominantely on
operator-theoretic
aspects of the subject. A series of minicourses formed the program's
center-
piece.
This volume consists of expository articles by participants in the
program
(plus collaborators, in two cases), including several articles based
on
minicourses. The opening article, by Donald Sarason, gives an
overview of
several aspects of the subject. The remaining articles, while more
special-
ized, are nevertheless designed in varying degrees to be accessible to
the
nonexpert. A range of topics is addressed: Bergman spaces (Hakan
Hedenmalm,
Karl Stroethoff); Hankel operators in various guises (Vladimir Peller,
Pamela Gorkin, Scott Saccone, Richard Rochberg); the Dirichlet space
(Zhijian
Wu); subnormal operators (John B. Conway and Liming Yang); operator
models
and related areas, especially interpolation problems and systems
theory
(Nikolai Nikolski and Vasily Vasyunin, Cora Sadosky, Nicholas Young,
Alexander
Kheifets, Harry Dym, James Rovnyak and coauthors). The concluding
article,
by Victor Vinnikov, describes an approach to certain commuting
families of
nonself-adjoint operators in which operator theory is linked with
algebraic
geometry.