# Monotone Matrix Functions and Analytic Continuation

Book

Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 207)

1. Front Matter
Pages i-ix
2. William F. Donoghue Jr.
Pages 1-17
3. William F. Donoghue Jr.
Pages 18-33
4. William F. Donoghue Jr.
Pages 34-41
5. William F. Donoghue Jr.
Pages 42-49
6. William F. Donoghue Jr.
Pages 50-62
7. William F. Donoghue Jr.
Pages 63-66
8. William F. Donoghue Jr.
Pages 67-77
9. William F. Donoghue Jr.
Pages 78-84
10. William F. Donoghue Jr.
Pages 85-87
11. William F. Donoghue Jr.
Pages 88-93
12. William F. Donoghue Jr.
Pages 94-99
13. William F. Donoghue Jr.
Pages 100-116
14. William F. Donoghue Jr.
Pages 117-127
15. William F. Donoghue Jr.
Pages 128-133
16. William F. Donoghue Jr.
Pages 134-139
17. William F. Donoghue Jr.
Pages 140-145
18. William F. Donoghue Jr.
Pages 146-153
19. William F. Donoghue Jr.
Pages 154-164
20. William F. Donoghue Jr.
Pages 165-169

### Introduction

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 trans­ parent as possible, we have adjoined short chapters containing the in­ formation about reproducing kernels, almost positive matrices and certain classes of conformal mappings needed for our proofs.

### Keywords

Analytische Funktion Monotone Funktion function proof spectral theorem theorem

#### Authors and affiliations

1. 1.University of CaliforniaIrvineUSA

### Bibliographic information

• Book Title Monotone Matrix Functions and Analytic Continuation
• Authors W.F.Jr. Donoghue
• Series Title Die Grundlehren der mathematischen Wissenschaften
• DOI https://doi.org/10.1007/978-3-642-65755-9
• Copyright Information Springer-Verlag Berlin Heidelberg 1974
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Hardcover ISBN 978-3-540-06543-2
• Softcover ISBN 978-3-642-65757-3
• eBook ISBN 978-3-642-65755-9
• Series ISSN 0072-7830
• Edition Number 1
• Number of Pages X, 184
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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