Pick Interpolation and Hilbert Function Spaces
This book is devoted to a study of the Pick interpolation problem via operator theory. The original problem was posed by G. Pick (professor of mathematics in Prague) in 1916 and was then studied independently in 1919 by Nevanlinna, apparently unaware of Pick’s work. A new look at the problem was taken in the 1960s by D. Sarason, a pioneer of the operator theory approach. He noticed that the Pick interpolation problem can be viewed as a question about the multiplier algebra of the Hardy space H2.
The exposition follows a one-semester course by the second author to an audience of PhD students and faculty staff. The book is geared so that it is accessible to graduate students interested in operator theory or spaces of holomorphic functions, and therefore contains background results as well as material based on research papers by the authors, which appears in book form for the first time. The approach taken here is not as straightforward as the classical one.
Three chapters (Chapters 2-4) build the necessary background on Hardy spaces and other indispensable spaces of holomorphic functions, and another one (Chapter 10) brings in standard results from operator theory and model theory. The pay-off is that the results obtained can be extensively generalised. The main results of the book include a proof that the Hardy space has the Pick property (with particular emphasis on the role of the positivity of the Pick matrix), a study of qualitative properties of the solutions to the Pick problem, a characterisation of spaces with the matrix-valued Pick property, a proof that Dirichlet and Sobolev spaces have the Pick property, the existence of a universal kernel with the Pick property, a study of the Pick problem in uniform algebras, the development of a hereditary functional calculus and its applications to a characterisation of operators that can be modelled by the adjoint of a given multiplication operator, and, a proof that the complete Pick property is equivalent to a certain localisation property for dilations.