Trajectory Spaces, Generalized Functions and Unbounded Operators
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1162 S.J. L,. van Eijndhoven J. de
Trajectory Spaces, Generalized Functions and Llnbounded Operators
Springer-Verlag Berlin Heidelberq New York Tokyo
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
1162 S.J. L,. van Eijndhoven J. de
Trajectory Spaces, Generalized Functions and Llnbounded Operators
Springer-Verlag Berlin Heidelberq New York Tokyo
Authors S.J.L. van Eijndhoven
J. de Graaf Eindhoven University of Technology Den Dolech 2, P.O. Box 513 5700 MB Eindhoven, The Netherlands
Mathematics Subject Classification (1980): 46A 12, 46F05, 46F 10,47030,81 B05 ISBN 3·540·16065·5 Sprinqer-Verlaq Berlin Heidelberg New York Tokyo ISBN 0·387·16065·5 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloqinq-m-Publication Data. Eijndhoven, Stephan us van, 1956- Trajectory Spaces, generalized functions, and unbounded operators. (Lecture notes in mathematics; 1162) Bibliography: p. Includes index. 1. Linear topological spaces. 2. Mappings (Mathematics) 3. Quantum theory. I. Graaf, Johannes de, 1942-. II. Title. III. Series: Lecture notes in mathematics (Springer-Verlag); 1162. 0A3.L28 no. 1162 [0A322] 510 s [515.7'3] 85-27810 ISBN 0-387-16065-5 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Sprinqer-Verlaq Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
Prologue I.
Analyticity spaces, trajectory spaces and linear mappings between them Introduction
10
SX,A The trajectory space TX,A Pairing and duality of SX,A and TX,A
11
22
1.4.
Continuous linear mappings between analyticity spaces and trajectory spaces
37
II.
Illustrative examples of analyticity spaces
1.11.2. 1.3.
The analyticity space
31
Introduction
45
11.1.
Analyticity spaces based on the Laplacian operator
47
11.2.
The GelfandShilov spaces
11.3.
Analyticity spaces related to classical polynomials
60
11.4.
Analyticity spaces related to unitary representations of Lie groups
71
III.
Compound spaces, tensor products and kernel theorems
sSa
Introduction
55
77
111.1. Compound spaces
78
111.2. The analyticitytrajectory space STZ;C,O
82
111.3. The trajectoryanalyticity space TSZ;C,O
98
111.4. Pairing and duality of STZ;C,O and TSZ;C,D
103
111.5. An inclusion diagram for compound spaces
108
II1.6. Topological tensor products and kernel theorems
116
Appendix
131
IV IV.
Algebras of continuous linear mappings on analyticity spaces and trajectory spaces Introduction
133
L(SX,A) IV.2. The algebra L(TX,A) IV.3. The algebra E(SX,A)
135
IV.1. The algebra
IV.4. Operator ideals in
149 162
L(SX,A) and L(TX,A)
17
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