Table Of ContentPOSITIVE SEMIGROUPS OF OPERATORS, AND APPLICATIONS
POSITIVE SEMIGROUPS
OF OPERATORS, AND
APPLICATIONS
Edited by
OLA BRATTELI
Mathematics Institute, University of Trondheim, Norway
and
PALLE E. T. J0RGENSEN
Dept. of Mathematics/El, University of Pennsylvania, Philadelphia, U.S.A.
Reprinted from
Acta Applicandae Mathematicae, Vol. 2, Nos. 3/4
~ D. Reidel Publishing Company
\fl[J
Dordrecht / Boston
ISBN-13: 978-94-009-6486-0 e-ISBN-13: 978-94-009-6484-6
DOl: 10.1007/978-94-009-6484-6
© 1984 by D. Reidel Publishing Company, Dordrecht, Holland
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ACT A APPLICANDAE MATHEMA TICAE
Volume 2 Nos. 3 and 4 Sept. IDee. 1984
POSITIVE SEMIGROUPS OF OPERATORS, AND APPLICATIONS
Edited by ala Bratteli and Palle E. T. JvrgeIJsell
OlA BRATTELI and PAllE E. T. J0RGENSEN / Positive Semigroups of
Operators, and Applications: Editors' Introduction 213
CHARLES J. K. BATTY and DEREK W. ROBINSON / Positive One-Parameter
Semigroups on Ordered Banach Spaces 221
W. KERSCHER and R. NAGEL / Asymptotic Behavior of One-Parameter Semi-
groups of Positive Operators 297
JURGEN VOIGT / Positivity in Time Dependent Linear Transport Theory 311
DA VID E. EVANS / Quantum Dynamical Semigroups, Symmetry Groups, and
locality 333
R. L. HUDSON and K. R. PARTHASARATHY / Stochastic Dilations of Uniformly
Continuous Completely Positive Semigroups 353
L. L. HELMS / Order Properties of Attractive Spin Systems 379
Book Reviews:
E. B. Davies: One-Parameter Semigroups (WILLIAM G. FARIS) 391
L. Asimow and A. J. Ellis: Convexity Theory and its Applications in
Functional Analysis (E. B. DAVIES) 395
Publications Received 399
Announcement 400
LIST OF FORTHCOMING PAPERS
FranQoise Brossier: Mathematical Modelisation of Equatorial Waves
John H. Cushman: Multiphase Transport Based on Compact Distributions
Philippe Destuynder: A Classification of Thin Shell Theories
Jerome A. Goldstein: Bound States and Scattered States for Contraction Semigroups
Minoru Kanehisa and Charles Delisi: Prediction of Protein and Nucleic Acid Structurt::
Problems and Prospects
V. V. Lychagin: Singularities of Multivalued Solutions of Nonlinear Differential Equa
hons, and Nonlinear Phenomena
Catherine A. Macken and Alan S. Perelson: Some Stochastic Models in Immunology
Servet Martinez A.: Non-Equilibrium Entropy on Stationary Markov Processes
Moshe Zakai: The Malliavin Calculus
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Acta Applicandae Mathematicae 2, 213-219. 0167-8019/84/0023-0213$01.05 213
© 1984 by D. Reidel Publishing Company.
Positive Semigroups of Operators, and
Applications: Editors' Introduction
OLA BRATTELI
Mathematics Institute. University of Trondheim. 7034 Trondheim. Norway
and
PALLE E. T. 10RGENSEN
Department of Mathematics/EI, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.
(Received: 28 February, 1984)
AMS (MOS) subject classifications (1980). Primary: 47B44, 47DOS, 47D07, 47D45, 47H06, 46H02, 20M20,
35K22; Secondary: 06F20, 26A33, 28DIO, 31A35, 31C12, 34C35, 34H05, 35110, 35P25, 41A35, 46A40,
47BS5, 47CIS, 53C20, S8GII, 58G20, 58G32, 60H99, 6013S, 60140, 60J45, 60160, 60165, 81E05, 82AIS,
82A70
Key words. One-parameter semigroups of operators, positivity, dissipative and dispersive operators,
diffusion processes, Markov fields, Poisson and Weierstrass transforms, Dirichlet problem, Schrodinger
equation, Riemannian manifold, Ricci curvature, Laplace operator, Fokker-Planck equation.
1. Introduction
In this collection of papers the editors have attempted to present a state of the art review
of the field of positive semigroups of operators with special emphasis on new (and
recent) research developments and applications.
The special issue begins with an article by C. J. K. Batty and D. W. Robinson on
positive one-parameter semigroups of operators on ordered Banach spaces. The first
half of this paper gives an updated treatment of ordered Banach spaces with many
illustrating examples and applications. Generation of positive semigroups, strict posi
tivity, irreducibility, and spectral properties (in parallel with the Perron-Frobenius
theory) are items from the second (main) part of the paper.
A wide interpretation of the positive semigroup concept would include all of Markov
processes. This, of course, is not our intention. Instead, we have made an effort to focus
on those newer mathematical theories which apply to classical and quantum statistical
mechanics. We regard the applications to operator algebras (i.e., the setting of quantum
models) as being particularly promising.
The Batty-Robinson paper is followed by five slightly more specialized articles
covering the following list of topics: asymptotic behavior, transport theory, quantum
*
Work supported in part by NSF.
214 OLA BRATIELI AND PALLE E. T. J0RGENSEN
dynamical semigroups, open systems and stochastics dilations, interacting particles,
and spin-systems.
We have encouraged an exposition of a discursive kind accessible to nonsuper
specialists.
The issue concludes with two long book reviews.
2. Historical Notes
The first major theorem on semigroups of operators was obtained in the 1940s by Hille
and Yosida (independently). We refer to the books [2, 3, 5, 11, 17, 19,21,29,31,34,
42, 51, 54 vol. 1] for background material. With vigorous activity in research and
applications up to the present day, the analytic theory of semigroups has, by now,
become a standard tool in probability, potential theory, harmonic analysis, partial
differential equations, functional analysis, ergodic theory, mathematical physics and,
more recently, biology and genetics.
Perhaps, it is even possible to notice different research trends in the field for each of
the post-war decades: (1) Diffusion processes, potential theory, and ergodic theory in
the Fifties with W. Feller, J. L. Doob, J. Deny, N. Dunford, E. Hille, S. Kakutani,
N. Kolmogorov, and K. Yosida; (2) parabolic partial differential equations and scattering
theory in the Sixties with A. Friedman, E. Nelson, E. Dynkin, G. Hunt, P. Lax,
J. Moser, R. Phillips, and T. Kato; (3) mathematical physics, optimal control theory,
and population processes in the Seventies with I. E. Segal, E. Nelson (the Markov
fields), J. Glimm, A. Jaffe; J. L. Lions, A. V. Balakrishnan; and T. G. Kurtz.
Mention should also be made of the connections to approximation theory (P. L.
Butzer, H. Berens, T. G. Kurtz), to classical infinite particle models (F. Spitzer, T. M.
Ligget, R. A. Holley, D. W. Stroock, L. Helms) and quantum models (D. Ruelle, D. W.
Robinson, H. Araki), to stochastic differential equations (K. Ita, H. P. McKean, T.
Hida), to nonlinear partial differential equations (I. E. Segal, F. Browder, M. Crandall,
A. Pazy, and T. Kato), to geometry of complete Riemannian manifolds (S.-T. Yau
[45,46]), and to constructive quantum field theory ([53,55]).
3. The Selection of Papers
We also mention work on hypercontractive semigroups, the infmite-dimensional Oren
stein -Uhlenbeck semigroups, and Schrodinger semigroups, by E. Nelson, I. E. Segal,
B. Simon, R. H0egh-Krohn, E. B. Davies, P. R. Chernoff, K. L. Chung, R. Carmona,
L. Gross, and others. While many of these specializations have been exposed in book
form, this is not so for the developments in the Eighties centering around positive
semigroups. This journal issue attempts to present a balanced exposition of these results
and applications. But even this rather recent research literature is large and scattered,
and it goes without mention that the topics which have been included represent a definite
choice on the part of the editors and authors. We apologize for any omissions. We feel,
however, that the present selection of papers is consistent and logically connected. But
there is certainly room for a different and disjoint selection of papers on related topics.
EDITORS' INTRODUCTION 215
4. Three Examples
In 1935, it was noticed by Hille that both the Poisson transform
f
(PJ) (x) = ~ 00 f(u + x) du (1)
n _ 00 U2 + y2
and the Weierstrass transform
f
(WJ)(x) = (nt)-1/2 _00 e-u2/, feu + x) du (2)
00
satisfy the semigroup law
S't + '2(f) = S't (S,,I), tl, t2 ~ O. (3)
In itself, a relatively simple fact which is now (and perhaps was also then) taken for
granted: In the same paper [18] which appears to have initiated the study of semigroups,
Hille also noted that (3) undoubtedly had been well-known earlier to mathematical
physicists since (PJ) (x) provides a solution to Dirichlet's problem for the upper
half-plane corresponding to the boundary function f(x) on the x-axis, whereas WJ is
a solution to the heat equation in a single dimension with initial temperature configu
ration f(x). The semigroup law (3) was noted as early as 1892 by P. Appell (J. de
Mathematique (4) 8, pp. 187-216). Hille also credited similar observations to E. Cesaro
(1902), and to G. Doetsch (1926).
The infinitesimal generator, A say, of a semigroup S, is defined as follows: It is
assumed that St acts in a linear space (which could consist of integrable, or square
integrable functions, bounded measurable, or continuous functions, just to mention a
few examples). A vector f (alias a function) is said to belong to the domain of the
generator A, and A(f) = h if the limit t-I(SJ - f) --+ h exists, as t --+ O.
Except for a few very special cases, A is only partially defmed, i.e., the domain of A
is not the whole space. Typically, it will not be a bounded operator relative to the norm
on the space under consideration. In the model examples (1) and (2) above, we have
the formulas,
A(f) = ~C:yf = 11" in case (2)
and
A(f) = l' in case (1),
where g(. ) denotes the M. Riesz conjugate function to g.
For the one-dimensional diffusion equation satisfied by the probability densities f(x),
the infmitesimal generator A is formally equal to a second-order differential operator
[ 12],
~ (~a(x)f
(Df) (x) = - b(X)f) + c(x)f(x) (4)
dx dx
216 OLA BRATIELI AND PALLE E. T. J0RGENSEN
where the functions a, a', b and c are defined in an open interval - OC! ~
'1 '2
< X < ~ 00, and a > O. The function c satisfies a certain technical condition [12].
(Actually, Feller reduces the analysis to the case c = 0.)
Since the semigroup S, is known to be determined uniquely by the generator A, the
analytic determination of S, (or equivalently the solution of the Cauchy problem)
therefore amounts to a specification of the domain for A. In his pioneering work [12],
Feller determined precisely the restrictions of the operator D in (4) which generate
semigroups of positive contraction (relative to the L I-norm) operators. Moreover, this
classification of the family of generators in terms of the domains (or the so-called
boundaries) had a direct interpretation in terms of the physical diffusion problem: the
regular boundary, the exit boundary, the entrance boundary, and the natural boundary
[12], Section 23. It made it possible for Feller to state the appropriate boundary
conditions for certain peculiar diffusion processes which had for years resisted all
attempts at a mathematical formulation.
5. Partial Differential Equations and Geometry
Here we wish to call attention to two separate early developments which appear to have
provided a particularly rich source of inspiration for later work, and which focused
attention on the above-mentioned domain question.
In 1954, Lax and Milgram [32] proved that if an alliptic operator P(x, alax) on an
n an,
open set in Rn is specified with Dirichlet boundary conditions on then the
resulting operator with this domain generates a semigroup on L 2(n).
The second result, due to Nelson in 1964 [39], provides a semigroup solution to the
Cauchy problem for the SchrOdinger equation (in any dimension) with a large class of
singular potentials, including the attractive 1/,2 potential. In the construction of S" the
infinitesimal generator A occurs as an extension of the Schrodinger operator
- (l/2m)A + V(x), or equivalently, a restriction of the corresponding maximal operator,
and the (time-irreversible) solution agrees, for Nelson's semigroup, with classical
mechanics in the correspondence limit.
Generalizing It6 [24], Yau constructed in [46], the diffusion semigroup S, which is
generated by a second-order elliptic operator L without a zero-order term on a specified
complete Riemannian manifold M. If L = A is the Laplacian of M, and if M has Ricci
curvature bounded from below, Yau's main theorem implies that the generator A of S,
is equal to the graph completion of A on the space of smooth functions with compact
supports on M (i.e., the minimal operator). Moreover, S" acting on bounded measur
able functions, maps the space Co(M) of continuous functions on M vanishing at infinity
into itself, S,(Co(M» c Co(M), t ~ O. As an application, Yau notes that, for a bounded
function f on M, the following two conditions are equivalent:
(i) Af is bounded in the distribution sense.
and
o.
(ii) t-I«SJ) (x) - f(x)) converges boundedly as t-+
EDITORS' INTRODUCTION 217
This means that semigroup theory may be applied directly to the study of the equation
I'!.f = h on M. In [45] Yau proves that, for h ~ 0, there are no nonconstant, nonnegative
solutions f in [j' for 1 < p < 00. From this, Yau gets the geometric fact that complete
noncom pact Riemannian manifolds with nonnegative Ricci curvature must have infinite
volume, a result which was announced earlier by Calabi [4].
6. Concluding Remarks
In several of the above results, positivity of the semigroup plays an important role. This
was also true, although only implicitly, for the early work of Hille and Yosida on the
Fokker-Planck equation, i.e., Equation (4) with c = O. But it was Phillips [41], and
Lumer and Phillips [37] who first called attention to the importance of dissipative and
dispersive properties of the generator in the context of linear operators in a Banach
space.
The generation theorems in the Batty-Robinson paper appear to be the most definitive
ones, so far, for this class of operators. The fundamental role played by the infinitesimal
operator, also for the understanding of order properties, in the commutative as well as
the noncommutative setting, are highlighted in a number of examples and applications
in the different papers, and it is hoped that this publication will be of interest to
researchers in a broad spectrum of the mathematical sub-divisions.
Acknowledgements
We are grateful to Professor Rhonda Hughes for reading the manuscripts and making
helpful suggestions.
References
An effort has been made to limit the size of this opening reference list, since the more specialized references
will be given in the individual papers to follow.
I. Balakrishnan, A. V.: Stochastic Differential Systems: Filtering and Control. A Function Space Approach,
Springer-Verlag, Berlin, Heidelberg, New York, 1973.
2. Balakrishnan, A. V.: Applied Functional Analysis, Application of Math. vol. 3, Springer-Verlag, Berlin,
Heidelberg, New York, 1976.
3. Butzer, P. L. and Berens, H.: Semi-Groups of Operators and Approximation, Die Grundlehren bd. 145,
Springer-Verlag, Berlin, Heidelberg, New York, 1967.
4. Calabi, E.: 'On Manifolds with Non-negative Ricci Curvature II', Notices Amer. Math. Soc. 22 (1975),
A205.
5. Davies, E. B.: One-parameter Semigroups, Academic Press, London New York, 1980.
6. Deny, 1.: Families fondamentales Noyaux associes', Ann. Inst. Fourier 3 (1951), 73-101.
7. Deny, 1.: 'Noyaux de convolution de Hunt et noyaux associes a une famille fondamentale', Ann. Inst.
Fourier 12 (1962), 643-667.
8. Doob, 1. L.: Stochastic Processes, Wiley, New York, 1953.
9. Doob, 1. L.: 'A Probability Approach to the Heat Equation', Trans. Amer. Math. Soc. 80 (1955),
216-280.
10. Dynkin, E. B.: 'Markoff Processes and Semigroups of Operators, and Infinitesimal Operators of
Markoff Processes', Teorya Veroyatn 1 (1956), 25-37.
