Partial Inner Product Spaces Theory and Applications
Partial Inner Product (PIP) Spaces are ubiquitous, e.g. Rigged Hilbert spaces, chains of Hilbert or Banach spaces (such as the Lebesgue spaces Lp over the real line), etc. In fact, most functional spaces used in (quantum) physics and in signal proces
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1986
Jean-Pierre Antoine · Camillo Trapani
Partial Inner Product Spaces Theory and Applications
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Jean-Pierre Antoine
Camillo Trapani
Institut de Physique Théorique Université catholique de Louvain 2, chemin du Cyclotron 1348 Louvain-la-Neuve Belgium [email protected]
Dipartimento di Matematica ed Applicazioni Università di Palermo Via Archirafi, 34 90123 Palermo Italy [email protected]
ISBN: 978-3-642-05135-7 e-ISBN: 978-3-642-05136-4 DOI 10.1007/978-3-642-05136-4 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009941068 Mathematics Subject Classification (2000): 46C50, 46Axx, 46Exx, 46Fxx, 47L60, 47Bxx, 81Qxx, 94A12 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Foreword
This volume has its origin in a longterm collaboration between Alex Grossmann (Marseille) and one of us (JPA), going back to 1967. This has resulted in a whole collection of notes, manuscripts, and joint papers. In particular, a large set of unpublished notes by AG (dubbed the ‘skeleton’) has proven extremely valuable for writing the book, and we thank him warmly for putting it at our disposal. JPA also thanks the Centre de Physique Th´eorique (CPT, Marseille) for its hearty hospitality at the time. Later on, almost thirty years ago, the two authors of this book started to interact (with a strong initial impulse of G. Epifanio, CT’s advisor at the time), mostly in the domain of partial operator algebras. This last collaboration has consisted entirely of bilateral visits between Louvain-la-Neuve and Palermo. We thank our home institutions for a constantly warm hospitality, as well as various funding agencies that made it possible, namely, the Commisariat G´en´eral aux Relations Internationales de la Communaut´e Fran¸caise de Belgique (Belgium), the Direzione Generale per le Relazioni Culturali del Ministero degli Affari Esteri Italiano and the Ministero dell’Universit` ae della Ricerca Scientifica (Italy). In the meantime, we also enjoyed the collaboration of many colleagues and students
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