Table Of ContentTexts and Readings in Mathematics 74
Kalyan Sinha
Sachi Srivastava
Theory of
Semigroups
and Applications
Texts and Readings in Mathematics
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Kalyan B. Sinha Sachi Srivastava
(cid:129)
Theory of Semigroups
and Applications
123
Kalyan B.Sinha SachiSrivastava
Jawaharlal Nehru Centrefor Advanced Department ofMathematics
ScientificResearch University of Delhi
Bangalore NewDelhi
India India
ISSN 2366-8725(electronic)
TextsandReadings inMathematics
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About the Authors
KalyanB.Sinha isprofessorandtheSERB-fellowattheJawaharlalNehruCentre
forAdvancedScientificResearch(JNCASR),andattheIndianInstituteofScience
(IISc), Bengaluru. Professor Sinha is an Indian mathematician who specialised in
mathematical theory of scattering, spectral theory of Schrödinger operators, and
quantum stochastic processes. He was awarded in 1988 the Shanti Swarup
BhatnagarPrizeforScienceandTechnology,thehighestscienceawardinIndia,in
the mathematical sciences category. A Fellow of the Indian Academy of Science
(IASc),Bengaluru,IndianNationalScienceAcademy(INSA),NewDelhi,andThe
World Academy of Sciences (TWAS), Italy, he completed his PhD from the
University of Rochester, New York, U.S.A.
Sachi Srivastava is associate professor at the Department of Mathematics,
UniversityofDelhi,India.SheobtainedherDPhildegreefromOxfordUniversity,
UKandtheMTechdegreefromtheUniversityofDelhi,India.Herareasofinterest
are functional analysis, operator theory, abstract differential equations, operator
algebras. She is also a life member of the American Mathematical Society and
Ramanujan Mathematical Society.
v
Preface
Semigroups (or groups, in many situations) of maps or operators
in a linear space have played important roles, mathematically en-
capsulating the idea of homogeneous evolution of many observed
systems, physical or otherwise. As an abstract mathematical disci-
pline, the theory of semigroups is fairly old, with the classical text,
Functional Analysis and Semigroups by Hille and Phillips [12] be-
ing probably the first one of its kind. Indeed, there have been a
good number of books and monographs on this topic written over
the years, many of which have been referred to in the present text.
Perhaps one of the reasons for having so many texts in this one
area of advanced mathematical analysis is the fact that the basic
theory of semigroups finds many applications in numerous areas of
enquiry: partial and ordinary differential equations, the theory of
probability and quantum and classical mechanics to name just a
few. In the present endeavour, along with the systematic develop-
ment of the subject, there is an emphasis on the explorations of
the contact areas and interfaces, supported by the presentations of
explicit computations, wherever feasible.
This book is aimed at the students in the masters level as well
as those in a doctoral programme in universities and research insti-
tutionsandenvisagesthepre-requisitesas:(i)agoodunderstanding
viii Preface
of real analysis with elements of the theory of measures and inte-
gration, (for example as in [23]), (ii) a first course in functional
analysis and in the theory of operators, say as in [5]. Many exam-
pleshave been givenineach chapter, partlytoinitiateandmotivate
the theory developed and partly to underscore the applications. As
mentioned earlier, several of these involve detailed analytical com-
putations, many of which have been undertaken in the text and
some others left as exercises. Instead of making a separate section
on exercises, they appear in line, in bold and in the relevant places
as the subject develops and the readers are encouraged to solve as
many of them as possible. It is suggested that a beginner may read
chapters 1 through 4 (except for sections 3.3 and 3.4) and leave the
rest for a second reading. In the Appendix we have collected some
standard results from the theory of unbounded operators, Fourier
transforms and Sobolev spaces which are required in our treatment
of the subject. It is worthwhile to bring to the attention of the
reader the fact that we have used the notation (cid:2)·,·(cid:3) to denote the
inner product in Hilbert spaces as well as to represent dual pair-
ing, and (cid:2)·,·(cid:3) will be taken to be linear in the left and conjugate
linear in the right entry. The present text arose out of the notes
of the lectures given by the first author (K. B. S.) – twice at the
Delhi Centre of the Indian Statistical Institute and once at the In-
dian Institute of Science, Bangalore and the interaction with the
students of those courses has helped shape the final product. Of
course, many existing texts on the subject have influenced the au-
thors and a particular mention needs to be made of the classical
treatise [12] and the books [11], [15] [19] and [27]. The monographs
[2] and [8] have also been referred to frequently. The authors re-
gret that the bibliography is far from exhaustive, instead they were
guided only by the need of the topics treated.
Preface ix
The choice of topics in this vastly developed subject is a diffi-
cult one and the authors have made an effort to stay closer to ap-
plications instead of bringing in too many abstract concepts. While
the chapters 2 and 3 make up the fundamentals of any discourse on
semigroup theory, the first chapter contains background material,
someofwhicharealsoofindependent interest. Chapter 4dealswith
the issue of the stability of classes of semigroups under small per-
turbationsaswellasthegeneralized strong continuity ofsemigroups
with respect to a parametric dependence. The chapters 5 and 6 deal
with special material, opening avenues for many applications: the
remarkable theorem of Chernoff leading to the Trotter-Kato prod-
uct formula which in turn motivates the Feynman-Kac formula for
a Schr¨odinger semigroup, and the Central Limit Theorem. Chap-
ter 6 deals with positivity-preserving (or semi-Markov) semigroups,
havingitsorigininthetheoryofprobabilityandconsidersperturba-
tions, not small in the sense of Chapter 4. The motivation for some
of the material in Chapter 5 and Chapter 6 comes from the theory
of probability and for an introduction to elements of that subject,
the reader may consult [18]. The last chapter gives a glimpse of how
the tools of the semigroup theory can be used to understand par-
tial differential operators in particular the wave and Schro¨dinger
operators.
The first author (K. B. S.) thanks the Indian Statistical In-
stitute, the Indian Institute of Science and most importantly the
Jawaharlal Nehru Centre for Advanced Scientific Research, Ban-
galore, for ready assistance, both direct and indirect, in making
this project a reality. He has special words of gratitude for the De-
partment of Science and Technology, Government of India, for the
SERB-Distinguished Fellowship, and for his wife Akhila for infinite
patience. The second author (S. S.) would like to acknowledge the
x Preface
support of the Department of Mathematics, University of Delhi in
this endeavour and of her husband, Manik. It is also a pleasure
to thank Tarachand Prajapati of the Department of Mathematics
at the University of Delhi for help, particularly with regards to the
drawingofthefigureinthebook.Lastbutnottheleast,theauthors
are grateful to the anonymous reviewer for many helpful comments
for the improvement in the presentation.
Kalyan B. Sinha Sachi Srivastava
Jawaharlal Nehru Centre for Department of Mathematics
Advanced Scientific Research University of Delhi
Bangalore Delhi
October 2016
