Positive Operators and Semigroups on Banach Lattices: Proceedings of a Caribbean Mathematics Foundation Conference 1990

150 Pages·1992·7.32 MB·English

by  Y. A. Abramovich, C. B. Huijsmans, C. D. Aliprantis (auth.), W. A. J. Luxemburg (eds.)

POSITIVE OPERATORS
AND SEMIGROUPS ON
BANACH LATTICES
Proceedings of a Caribbean
Mathematics Foundation
Conference 1990
Edited by
C. B. HUIJSMANS
Department of Mathematics and
Computer Science,
Leiden University, The Netherlands
and
W. A. J. LUXEMBURG
California Institute a/Technology,
Pasadena. U.S.A.
Reprinted from
Acta Applicandae Mathematicae, Vol. 27, Nos. 1-2 (1992)
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
?osltlve operators and semlgroups on Banach lattlces proceedlngs of
the Carlbbean Mathematlcs Foundatl0n's conference. 1990 / edlted by
C.S. HU1Jsmans and W.A.J. Luxemburg.
p. cm.
ISBN 978-90-481-4205-7 ISBN 978-94-017-2721-1 (eBook)
DOI 10.1007/978-94-017-2721-1
1. Posltlve operators--Congresses. 2. Semlgroups of opera tors-
-Corgresses. 3. Banach lattlces--Congresses. I. HU1Jsmans. C. B.
II. Luxemburg. W. A. J .• 1929- III. Carlbbean Mathematlcs
Foundatl0n.
QA329.2.P67 1992
5'5' .7242--dc20 92-26747
ISIJN 978-90-481-4205-7
Printed on acid-free paper
Ali Rights Reserved
© 1992 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1992
No part of the material protected by this copyright notice may be reproduced Of
utilized in any fonn Of by any means, electronic Of mechanical,
includ ing photocopying, recording Of by any infonnation storage and
retrieval system, without written pennission from the copyright owner.
Table of Contents
Preface v
List of Participants VB
Y. A. ABRAMOVICH, C. D. ALIPRANTIS, and O. BURKINSHAW / Positive
Operators on Krein Spaces
Y. A. ABRAMOVICH and W. FILTER / A Remark on the Representation of
Vector Lattices as Spaces of Continuous Real-Valued Functions 23
W. ARENDT and J. VOIGT / Domination of Uniformly Continuous Semigroups 27
S. J. BERNAU / Sums and Extensions of Vector Lattice Homomorphisms 33
B. EBERHARDT and G. GREINER / Baillon's Theorem on Maximal Regularity 47
A. W. HAGER and J. MARTINEZ / Fraction-Dense Algebras and Spaces 55
C. B. HUIJSMANS and W. A. J. LUXEMBURG / An Alternative Proof of a
Radon-Nikodym Theorem for Lattice Homomorphisms 67
C. B. HUIJSMANS and B. DE PAGTER / Some Remarks on Disjointness
Preserving Operators 73
L. MALIGRANDA / Weakly Compact Operators and Interpolation 79
P. MEYER-NIEBERG / Aspects of Local Spectral Theory for Positive Operators 91
B. DE PAGTER / A Wiener-Young Type Theorem for Dual Semigroups 101
A. R. SCHEP / Krivine' s Theorem and Indices of a Banach Lattice 111
A. W. WICKSTEAD I Representations of Archimedean Riesz Spaces by Con-
tinuous Functions 123
X.-D. ZHANG / Some Aspects of the Spectral Theory of Positive Operators 135
Problem Section 143
Acta Applicandae Mathematicae 27: v-vi, 1992. v
© 1992 Kluwer Academic Publishers.
Preface
During the last thirty years advances in the theory of ordered algebraic struc-
tures such as vector lattices (Riesz spaces), f -algebras and Banach lattices have
played a very important role in the recent development of the theory of positive
linear operators that has its roots in the fundamental results of Frobenius and
Perron about the spectral properties of positive matrices. Moreover, motivated
by problems concerning partial differential equations, particularly those dealing
with initial value problems, probability theory (Markov processes), mathematical
physics and control theory, the theory of one-parameter semi groups of positive
linear operators on' Banach lattices has undergone a tremendous growth during
the last decades.
From June 18 through June 22, 1990 on the Caribbean island of Cura<;ao
(Netherlands Antilles) a small workshop was held devoted to the theory of pos-
itive operators and their semigroups. Following the workshop a conference was
held from June 25-June 29 primarily on recent advances in this area of positive
operators. The workshop and the conference took place under the auspices of
the Caribbean Mathematical Foundation (CMF) under the directorship of Dr J.
Martinez.
The purpose of the workshop, conducted by C.B. Huijsmans, W.AJ. Luxem-
burg and B. de Pagter, was to present to a group of interested mathematicians
from the Caribbean and Latin America an up-to-date account of the main re-
sults of the theory of positive operators on Banach lattices. The workshop was
attended by mathematicians from Florida U.S.A., Guyana, Panama, Surinam and
Venezuela. There were three one and a half hours sessions per day during five
days.
The conference following the workshop was organised by C.B. Huijsmans and
W.A.J. Luxemburg with the main purpose to bring together a group of likeminded
specialists from the U.S.A. and Western Europe to present their recent results
and to discuss their research interests. The worked-out versions of the papers that
were presented at the conference and some related contributions are collected in
these proceedings. All the submitted articles were refereed.
We take this opportunity to thank all the participants of both the workshop and
the conference for their contributions which made this mathematical gathering
so successful.
The financial contribution by CMF (supported by NSF, the University of
VI PREFACE
Florida, de Universiteit van de Nederlandse Antillen, Fondo Fundashon Univer-
sidat and het Bestuurscollege Eilandsgebied Curas,:ao) is greatly appreciated by
all the participants. Notably, we wish to express our sincere gratitude to the di-
rector of the CMF, Dr J. Martinez, for his encouragement, continuous support
and participation. Without his unceasing efforts this workshop and conference
would never have taken place. Finally, we thank the editorial staff of Kluwer
Academic Publishers for their cooperation and support during the preparation of
this manuscript.
Leiden/pasadena, Summer 1992 C.B. HUIJSMANS, W.A.J. LUXEMBURG
Acta Applicandae Mathematicae 27: 1992. Vll
List of Participants
C.D. ALIPRANTIS, Department of Mathematics, IUPUI, 1125 East 38th Street,
Indianapolis, Indiana 46205-2810, U.S.A.
W. ARENDT, Equipe de MatMmatiques, Universite de Franche-Comte, 25030
Besan<;:on Cedex, France.
SJ. BERNAU, Department of Mathematical Sciences, The University of Texas
at EI Paso, EI Paso, Texas, 79968-0514, U.S.A.
G. GREINER, Mathematisches Institut, UniversiHit Tiibingen, Auf der Morgen-
stelle 10, D-4700 Tiibingen 1, Germany.
C.B. HUIJSMANS, Mathematisch Instituut, Rijksuniversiteit Leiden, P.O. Box
9512, Niels Bohrweg 1,2300 RA Leiden, The Netherlands.
W.A.J. LUXEMBURG, Department of Mathematics 253-37, California Institute
of Technology, Pasadena, California 91125, U.S.A.
L. MALIGRANDA, Departamento de Matematicas, IVIC, Apartado 21827,
Cara<;:as 1020-A, Venezuela.
J. MARTINEZ, Department of Mathematics, University of Florida, 201 Walker
Hall, Gainesville, Florida 32611-2082, U.S.A.
P. MEYER-N lEBERG, Fachbereich Mathematik, Universitat Osnabriick, Albrecht-
straSe 28,4500 Osnabrock, Postfach 4469, Germany.
B. de PAGTER, Faculteit der Wiskunde en Informatica, Mekelweg 4, Postbus
5031,2600 GA Delft, The Netherlands.
A.R. SCHEP, Department of Mathematics, University of South Carolina, Colum-
bia, South Carolina 29208, U.S.A.
A.W. WICKSTEAD, Department of Pure Mathematics, The Queen's University
of Belfast, Belfast BTI INN, Northern Ireland, u.K.
X-D. ZHANG, Department of Mathematics 253-37, California Institute of
Technology, Pasadena, California 91125, U.S.A.
Acta Applicandae Mathematicae 27: 1-22, 1992.
© 1992 Kluwer Academic Publishers.
Positi ve Operators on Krein Spaces
Dedicated to the memory of M.G. Krein (1907-1989)
Y.A. ABRAMOVICH, C.D. ALIPRANTIS and O. BURKINS HAW
Department of Mathematics, IUPUl, Indianapolis, IN 46205, U.S.A
(Received: 27 April 1992)
Abstract. A Krein operator is a positive operator, acting on a partially ordered Banach space,
that carries positive elements to strong units. The purpose of this paper is to present a survey
of the remarkable spectral properties (most of which were established by M.G. Krein) of these
operators. The proofs presented here seem to be simpler than the ones existing in the literatnre.
Some new results are also obtained. For instance, it is shown that every positive operator on a
Krein space which is not a multiple of the identity operator has a nontrivial hyperinvariant subspace.
Mathematics Subject Classifications (1991): 46C50, 47B65, 47B37
Key words: partially ordered Banach space, Krein space, Krein operator, hyperinvariant subspace
1. Introduction
The classical theorem of M.G. Krein and M.A. Rutman asserting that the spectral
radius of a nonquasinilpotent positive compact operator is an eigenvalue having
a positive eigenvector is well known and widely used in the literature. This
result was just one of the many important results concerning spectral properties
of positive operators that appeared in [8]. It seems that most of these results
are not well known and their proofs are not readily available. The purpose of
this paper is to present a survey (with proofs) of these results in the setting of
partially ordered Banach spaces. This gives us the opportunity to present at the
same time some of the basic order properties of partially ordered normed spaces.
Due to our advantage of hindsight, many of our proofs are simplified versions
of existing proofs in the literature and are presented in a systematic way using
modern techniques and terminology.
We are especially interested in the intriguing class of operators that carry
positive elements to order units. We shall call these operators Krein operators.
Some remarkable spectral properties of Krein operators are discussed here and a
variety of examples illustrate their usefulness and importance.
In general, the paper presents results that deal with eigenvalues and fixed
2 Y.A. ABRAMOVICH, C.D. ALIPRANTIS AND O. BURKINS HAW
points of positive operators (or families of positive operators) or their adjoints.
Most of these beautiful results are due to either M.G. Krein and his collaborators
or they have been influenced by his work. Accordingly, we dedicate the paper
to the memory of this great mathematician.
2. Partially Ordered Normed Spaces
Recall that a real vector space X equipped with a partial order 2: is said to be
a partially ordered vector space or simply an ordered vector space whenever
1) x 2: y implies ax 2: ay for all a 2: 0; and
2) x ~ y implies x + z ~ y + z for all Z E X.
The positive cone (or simply the cone) X+ of X is the set of all positive elements
of X, i.e., X+ = {x E X: x ~ O}. The cone X+ is said to be generating
whenever X = X+ - X+, i.e., whenever every vector can be written as a
difference of two positive vectors.
A partially ordered vector space is said to be Archimedean whenever nx ::; y
for each n and some x, y imply x ::; O. Notice that in an Archimedean partially
ordered vector space y 2: 0 and -EY ::; X ::; Ey for all E > 0 imply x = O.
LEMMA 2.1 If a partially ordered vector space X admits a Hausdoif.! linear
topology for which X+ is closed, then X is Archimedean.
Proof. Let l' be a Hausdorff linear topology on a partially ordered vector
space X such that X+ is T-closed. Now assume nx ::; y for all n and some
x, y E X. Then *y - x 2: 0 and *y - x ~ -x imply -x 2: 0 or x ::; o. Q.E.D.
THEOREM 2.2 (M.G. Krein-Smulian) If X is a Banach space ordered by a
closed generating cone, then there exists a constant J'v1 > 0 such that for each
x E X there are vectors Xl, X2 E X+ satisfying
x = Xl - X2 and Ilxill ::; Mllxll (i = 1,2).
Proof. The proof below is due to B.Z. Vulikh [15] and closely resembles the
proof of the open mapping theorem. For each real number t 2: 0 we define the
set
Et = {x EX: :3Xl,X2 E X+ with x = Xl - X2 and Ilxill ::; t (i = 1,2)}.
Clearly, each E t is convex, symmetric, and 0 E E t. In addition, note that aEt =
Eat holds for each a 2: 0, and 0 ::; s ::; t implies Es S;;; E t .
Since X+ is a generating cone, we see that X = U~=l En. So, by the Baire
Category Theorem, there exist a natural number k, an Xo E X and an r > 0 such
that the closed ball C(xo, r) satisfies
C(xo, r) = {x EX: Ilxo - xii::; r} S;;; Ek.
POSITIVE OPERATORS ON KREIN SPACES 3
From the symmetry of Ek we see that Ilxll ::::; r implies Xo + x, x - Xo E Ek.
and so by the convexity of Ek we infer that x = ~(xo + x) + ~(x - xo) E Ek.
Therefore, C(O, r) ~ Ek .
We claim that C(O, r) ~ E2k holds. If this is established, then clearly for
x E X there exist XI,X2 E X+ with x = XI-X2 and Ilxill ::::; 2:llxll for i = 1,2.
To establish that C(O, r) ~ E2k holds, let x E C(O, r). Then ~x E Ek. So
there exists an Xl E Ek such that II~x - XIII < i. Note that IlxI11 ::::; rand
Ilx - xIII = II (~X - Xl) + ~xll < ~ + ~ = &.
So, X-Xl E C (0, 1) = !C(O, r) ~ Ef£. Repeating the same argument and using
2
X - Xl instead of x, we see that there exists an X2 E E!:£ such that IIx211 :; 1
2
and Ilx - Xl - x211 < i. So, by an inductive argument, we see that there exists
a sequence {xn} in X such that
1) Xn E E_k_ and Ilxnll ::::; 2LI for n = 1,2, ... ; and
2n - 1
2) Ilx - I:~l xiii < {n.
Now for each n choose Yn, Zn E X+ such that Xn = Yn - Zn, IIYnl1 ::::; 2nk_I'
and Ilznll ::::; 2nk_ l . If Y = I:~=l Yn and Z = I:~=l Zn, then (since X+ is norm
closed) y, Z E X+. From I:~l IIYnl1 ::::; 2k and I:~=l Ilznll ::::; 2k we see that
Y - Z E E2k. Finally, from 2) we get X = I:~l Xi = Y - Z E E2k. and the proof
is finished. Q.E.D.
COROLLARY 2.3 Let X be a Banach space partially ordered by a closed gen-
erating cone. If Xn -----+ X holds in X, then there exist two sequences {Yn} and
{zn} in X+ satisfying
1) Yn -----+ Y and Zn -----+ z;
2) Xn = Yn - Zn for each n; and
3) X = Y - z.
Proof Using Theorem 2.2, we know that there exist an M > °a nd two
sequences {an} and {bn} in X+ satisfying Xn -x = an -bn, Ilanll ::::; Mllxn -xii,
and Ilbnll ::::; Mllxn- xii for each n. So, an -+ °a nd bn -+ 0. Now write X = y-z
with y, Z E X+, then let Yn = y+an and Zn = z+bn and note that the sequences
{Yn} and {zn} satisfy the desired properties. Q.E.D.
COROLLARY 2.4 Let X be a Banach space partially ordered by a closed gener-
ating cone and let Y be a topological vector space. Then an operator T: X -----+ Y
is continuous if and only if T: X+ -----+ Y is continuous.
An operator T: X -----+ Y between two ordered vector spaces (where, as usual,
'operator' means 'linear operator') is said to be positive whenever X ?: °i mplies
T(x) ?: 0.
4 Y.A. ABRAMOVICH, C.O. ALIPRANTIS AND O. BURKINSHAW
It is remarkable that quite often positive operators are automatically continu-
ous. This was first proved by M.G. Krein for positive linear functionals [8] and
later was generalized in several contexts by various authors; see, for instance,
[3], [10],[11], and [14]. The next result, due to G.Ya. Lozanovsky, is the strongest
in this direction and appeared in [16].
COROLLARY 2.5 (Lozanovsky) Let X and Y be two partially ordered Banach
spaces with closed cones. If the cone of X is also generating, then every positive
operator T: X -+ Y is continuous.
Proof It suffices to show that the operator T has a closed graph. So, assume
Xn -+ 0 in X and TXn -+ Y in Y. By passing to a subsequence, we can also
assume that L~=l nllxnll < 00. By Theorem 2.2 there exist an M > 0 and two
sequences {Yn} and {zn} in X+ satisfying Xn = Yn - Zn, IIYnl1 :s: Mllxnll, and
Ilznll :s: Mllxnll for each n. Since X+ is closed, the vector Z = L~=l n(Yn +zn)
in X belongs to X+, and -z :s: nXn :s: Z holds for each n. From the positivity
of T we infer that -~Tz :s: TXn :s: ~Tz. Using that Y+ is also closed, we
conclude that 0 S y :s: O. That is, y = 0 and the proof is finished. Q.E.D.
3. Krein Spaces
Let X be a partially ordered vector space. A vector u E X+ is said to be a strong
unit (or simply a unit) whenever for each x E X there exists an a > 0 such that
x S au. The set of all units in X will be denoted by U. Clearly,
and aU = U for all a > O.
If an ordered vector space has a unit, then it is clear that its cone is generating.
DEFINITION 3.1 A partially ordered Banach space X is said to be a Krein
space whenever
a) X+ is closed; and
b) X has strong units, i.e., U ic 0.
Notice that every Krein space is automatically Archimedean and its cone is
generating. Since the cone of a Krein space is closed, its order intervals are
likewise closed sets. Here are some examples of Krein spaces.
The classical Banach lattices C(K) of all real-valued continuous functions
on a Hausdorff compact topological space K. The algebraic, order, and
lattice operations are defined pointwise and the norm is the sup norm. The
constant function 1 is a unit. Incidentally, the reader can convince himself
that the spaces C(K) are the only Banach lattices that are Krein spaces.
The vector space Ck[a, b] of all k-times continuously differentiable real-
valued functions on a (bounded) closed interval [a, b]. The algebraic and
order operations are defined pointwise. The norm is defined by
Ilxll = Ilxlloo + Ilx'lloo + ... + Ilx(k) 1100'