Operator Analysis: Hilbert Space Methods in Complex Analysis: 219 (Cambridge Tracts in Mathematics, Series Number 219)

This book shows how operator theory interacts with function theory in one and several variables. The authors develop the theory in detail, leading the reader to the cutting edge of contemporary research. It starts with a treatment of the theory of bounded holomorphic functions on the unit disc. Model theory and the network realization formula are used to solve Nevanlinna-Pick interpolation problems, and the same techniques are shown to work on the bidisc, the symmetrized bidisc, and other domains. The techniques are powerful enough to prove the Julia-Carathéodory theorem on the bidisc, Lempert's theorem on invariant metrics in convex domains, the Oka extension theorem, and to generalize Loewner's matrix monotonicity results to several variables. In Part II, the book gives an introduction to non-commutative function theory, and shows how model theory and the network realization formula can be used to understand functions of non-commuting matrices.

Jim Agler is Distinguished Professor Emeritus at the University of California, San Diego. He received the G. de B. Robinson award from the Canadian Mathematical Society in 2016 and delivered the 2017 London Mathematical Society Invited Lectures. He is the co-author of Pick Interpolation and Hilbert Function Spaces (2002).

John Edward McCarthy is the Spencer T. Olin Professor of Arts and Sciences at Washington University, St Louis, and chair of the Department of Mathematics and Statistics. He received the G. de B. Robinson award from the Canadian Mathematical Society (2016) and was co-author of Pick Interpolation and Hilbert Function Spaces (2002).

Nicholas John Young is Research Professor at Leeds University and Senior Research Investigator at University of Newcastle upon Tyne. He is the author of An Introduction to Hilbert Space (Cambridge, 1988) and approximately 100 research articles in analysis.