# Monotone Matrix Functions and Analytic Continuation

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

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

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.

Analytische Funktion Monotone Funktion function proof spectral theorem theorem

- 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 Springer Book Archive
- Print ISBN 978-3-642-65757-3
- Online ISBN 978-3-642-65755-9
- Series Print ISSN 0072-7830
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