Extension of Positive Operators and Korovkin Theorems
- PDF / 8,998,977 Bytes
- 195 Pages / 461 x 684 pts Page_size
- 73 Downloads / 192 Views
904 Klaus Donner
Extension of Positive Operators and Korovkin Theorems ETHICS ETH-BIB
III 1111111111111 00100000802840
Springer-Verlag Berlin Heidelberg New York 1982
Authors
Klaus Donner Mathematisches Institut, Universitat Eriangen-NOrnberg BismarckstraBe 11/2, 8520 Erlangen
AMS Subject Classifications (1980) 40-A-05, 46-A-22, 46-B-30
ISBN 3-540-11183-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11183-2 Springer-Verlag New York Heidelberg Berlin
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 Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Contents
iv
Introduction Notations. Section 1 : Cone imbeddings for vector lattices
2
Section 2 : A vector-valued Hahn-Banach theorem
12
Section 3 : Bisublinear and subbilinear functionals
30
Section 4 : Extensions of L
1-valued
positive operators.
Section 5: Extension of positive operators in LP-spaces.
68 84
Section 6 : The Korovkin closure for equicontinuous nets of positive operators . .
" ....
105
Section 7: Korovkin theorems for the identity mapping on classical Banach lattices.
127
Section 8: Convergence to vector lattice homomorphisms and essential sets.
162
List of symbols.
174
Literature
177
Index . . .
182
Introduction
Examining the various branches of functional analysis, it will be difficult to find a section in which extension problems of linear operators are completely absent. Usually, there are certain properties such as continuity, positivity, compactness, contractivity, or commutativity with given operators that have to be preserved in connection with the extension of the linear operator in question. For linear functionals, extension problems can often be satisfactorily settled using the HahnBanach theorem and its various consequences. Extending linear operators, however, turns out to be a troublesome enterprise. Analysing the proofs of the known extension theorems for linear mappings we corne up with two principal arguments: 1) Continuous linear operators defined on dense subspaces possess a continuous linear extension to the whole space. 2)
If H is a linear subspace of an arbitrary real vector space E and F is a Dedekind complete vector lattice, then a linear operator T : H
t
F dominated by a sublinear mapping P : E
to a linear operator
T
t
F can be extended
defined on E under the domination of P.
(The
vectorvalued version of the HahnBanach theorem). It is not intended to offer a survey of the numerous results based on these two theorems. On the other hand, it soon becomes evident that the majority of the extension problems fo
Data Loading...
