Math 523, Fall 2008

Bounded Analytic Functions

Instructor          John E. M^cCarthy
Class                  TuTh 10.00-11.30 in Cupples I, 111
Office                 105 Cupples I
Office Hours      M 3:00-4:00, Tu 2:30-3:30, Th: 3:00-4:00, or by appointment
Phone                 935-6753

Important Announcement: There will be no class on Thursday August 28^th. The first class will be on Tuesday September 2^nd.

Prerequisites

Complex Analysis 5021-5022 and Real Analysis 5051-5052

Content

We shall discuss bounded analytic functions, and the Hilbert spaces on which they act as multipliers. Our primary focus will be on the disk, where the most important Hilbert space is the Hardy space, and the algebra of bounded analytic functions, called H∞, is best understood (though many mysteries remain). However, we shall also foray into other domains, both in one and more dimensions. The topics we shall cover include: Blaschke products, interpolation, Toeplitz operators, Hankel operators, interpolating sequences, and Carleson’s corona theorem.

Basis for Grading

Grading will be trivial provided you participate in class.

Bibliography

The classic text is John Garnett’s “Bounded Analytic Functions”. This is a terrific book, and if you understand everything in it, you are ready to receive your Ph.D. My perspective is a bit more operator-oriented, as in Ron Douglas’s “Banach algebra techniques in Operator Theory”, or my book with Jim Agler “Pick interpolation and Hilbert function spaces”.

Books that focus on the Hardy space include Peter Duren’s “Theory of H^p spaces” and Paul Koosis’s “Introduction to H^p spaces”. A Banach algebra approach is in Kenneth Hoffman’s “Banach spaces of analytic functions”. A book that has a trove of information and elegant proofs, but is hard to read if this is your first time through the material, is Nikolai Nikolskii’s “Treatise on the Shift Operator”.