A Pick function is a function that is analytic in the upper half-plane with positive imaginary part. In the first part of this book we try to give a readable account of this class of functions as well as one of the standard proofs of the spectral theorem based on properties of this class. In the remainder of the book we treat a closely related topic: Loewner's theory of monotone matrix functions and his analytic continuation theorem which guarantees that a real function on an interval of the real axis which is a monotone matrix function of arbitrarily high order is the restriction to that interval of a Pick function. In recent years this theorem has been complemented by the Loewner-FitzGerald theorem, giving necessary and sufficient conditions that the continuation provided by Loewner's theorem be univalent. In order that our presentation should be as complete and transยญ parent as possible, we have adjoined short chapters containing the inยญ formation about reproducing kernels, almost positive matrices and certain classes of conformal mappings needed for our proofs
CONTENT
I. Preliminaries -- II. Pick Functions -- III. Pick Matrices and Loewner Determinants -- IV. Fatou Theorems -- V. The Spectral Theorem -- VI. One-Dimensional Perturbations -- VII. Monotone Matrix Functions -- VIII. Sufficient Conditions -- IX. Loewnerโ{128}{153}s Theorem -- X. Reproducing Kernels -- XI. Nagy-Koranyi Proof of Loewnerโ{128}{153}s Theorem -- XII. The Cauchy Interpolation Problem -- XIII. Interpolation by Pick Functions -- XIV. The Interpolation of Monotone Matrix Functions -- XV. Almost Positive Matrices -- XVI. The Analytic Continuation of Bergman Kernels -- XVII. The Loewner-FitzGerald Theorem -- XVIII. Loewnerโ{128}{153}s Differential Equation -- XIX. More Analytic Continuation -- Notes and Comment
