Springer Proceedings in Mathematics & Statistics Jacek Banasiak Adam Bobrowski Mirosław Lachowicz Editors Semigroups of Operators - Theory and Applications Będlewo, Poland, October 2013 Springer Proceedings in Mathematics & Statistics Volume 113 Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluationoftheinterest,scientificquality,andtimelinessofeachproposalatthe handsofthepublisher,individualcontributionsareallrefereedtothehighquality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today. More information about this series at http://www.springer.com/series/10533 Jacek Banasiak Adam Bobrowski (cid:129) ł Miros aw Lachowicz Editors Semigroups of Operators - Theory and Applications ę B dlewo, Poland, October 2013 123 Editors Jacek Banasiak Adam Bobrowski School ofMathematics, Statistics LublinUniversityof Technology andComputer Science Lublin Universityof KwaZulu-Natal Poland Durban South Africa MirosławLachowicz Instituteof AppliedMathematics and andMechanics Universityof Warsaw Instituteof Mathematics Warsaw Technical University ofŁódź Poland Łódź Poland ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings inMathematics &Statistics ISBN 978-3-319-12144-4 ISBN 978-3-319-12145-1 (eBook) DOI 10.1007/978-3-319-12145-1 LibraryofCongressControlNumber:2014956197 MathematicsSubjectClassification(2010):34-XX,45-XX,46-XX,47-XX,49-XX,60-XX,92-XX SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) The volume is dedicated to Prof. Jan Kisyński on the occasion of his 80th birthday Preface The foundations of the theory of semigroups of operators were established in the first half of the twentieth century as a formalization of the view emphasized by A. Hadamard that an autonomous deterministic system is described by a one- parameter semigroup of transformations. Hence, the theory was first designed as a functionalanalyticlanguageofpartialdifferentialequationsbutsoonalsoprovedto be an important tool in stochastic processes, involving mathematicians like W. Feller in developing its basics. One could say that the theory of semigroups reached its maturity in the fourth and fifth decades of the previous century, when the major generation theorems were established thanks to the work of K. Yosida, E.Hille,R.S.Phillips,I.Miyadera,and,ofcourseW.Feller.Itattaineditsfirstapex with the publication of the famous book Functional Analysis and Semi-Groups by E. Hille and R.S. Phillips, which reflected its predominantly functional-analytic character.Sincethen,semigrouptheoryhasbeenrapidlydevelopingasatheory“in itself”,posingnewandfascinatinginternalquestions,butalsoithasbeenreaching out to applications ranging from the classical ones such as partial differential equations and stochastic processes to less standard such as integro-differential and functional-differential equations, quantum mechanics, population biology, or con- trol theory. Furthermore, though inherently linear and autonomous, semigroup theory proved indispensable in describing both nonlinear and nonautonomous evolutionary phenomena. Thanks to intensive research, the theory reached some stateofperfectioninthelastdecadesofthepreviouscenturyandthewealthofthe amassed results, both theoretical and applied, has been reflected in a series of excellent books presenting the state of the art in the field and complementing the fundamental treatise of E. Hille and R.S. Phillips. These include monographs by E.B. Davies, A. Belleni-Morante, A. Pazy, J.A. Goldstein, and K.J. Engel and R. Nagel, which present the functional analytic point of view, and by S.N. Ethier and T.G. Kurtz, W. Feller and E.B. Dynkin, which are instrumental in under- standing the role of semigroups in Markov processes. These authors were largely responsible for bringing the theory to the current state and we are pleased to note thatasurveypaperofJ.A.GoldsteinandR.Nagelisoneofthecornerstonesofthe presented volume. vii viii Preface In the opinion of the editors of this volume, the recent years have witnessed an even increasing intertwining of applied and pure aspects of the theory. Semigroup methods, often in competition with other approaches, have been successfully appliedinconcreteproblemsencountered,e.g.,invariousbranchesofmathematical biology (including population dynamics, population genetics, coagulation and fragmentationprocesses,signalingpathways,etc.)which,inturn,haveinspiredthe development of new mathematical methods and important discoveries in the field. At the same time, despite the fact that semigroup theory was considered to be almostcompleteinthe1980s,itscorehasbeencontinuallyenlargedfarbeyondthe expectation to include, for instance, the theory of positive semigroups, long-term behavior of semigroups (including chaotic dynamics), systematic treatment of boundaryconditions,orgenerationtheoremsforcosinefamilies,whichhavefound important applications. It has been, however, felt that the exchange of the results between various communitiesthatusesemigroupmethodshavebeentooslow,possiblyduetooften hermetic languages used in particular applications. The conference was therefore meant as a forum for mathematicians using the theory of semigroups to meet and talk about the problems they have been working on. For those who were working on its pure aspect, this was a chance to present their theoretical results and learn of the developments and challenges in applied branches. And vice versa—those working on applied problems were able to put their work in the more general context and learn of the new discoveries in other fields. The fact that at the same timewecouldcelebratethe80thbirthdayofProf.JanKisyński,whohasplayedan essential role in the development of the theory and bringing together its probabi- listic and analytic aspects, gave the conference another dimension. The conference brought together 107 participants from 16 countries, who pre- sented five plenary and 86 contributed talks. This volume contains 21 talks that have undergone strict refereeing process before being accepted. While not pre- tending to have covered any significant part of the field, the editors nevertheless believethatthepresentedselectionwillgivethereaderaglimpseofthisfascinating area of research, from a classical survey of the history of the theory and its applications, through the discussion of some pertinent theoretical questions, to applications in control theory, financial models, quantum mechanics and other applied disciplines. The editors, who were also theconvenersoftheconference, aregrateful forthe support without which the conference would not have been possible. We received financial support from the International Banach Centre, the Mathematical Institute of the Polish Academy of Sciences, Lublin University of Technology, Łódź Uni- versity of Technology, the University of Warsaw, the National Science Centre Grants NCN N201605640 and NCN 6081/B/H03/2011/40, and the National Preface ix ResearchFoundationofSouthAfrica.GreatthanksgotothemembersoftheLocal Organizing Committee, Małgorzata Murat, Adam Gregosiewicz, and Zbigniew Łagodowski. Finally, we thank the organizers of the Special Sessions, plenary speakers, and all participants. Durban, Łódź, August 2014 Jacek Banasiak Lublin Adam Bobrowski Warsaw Mirosław Lachowicz Contents Part I Theory The Evolution of Operator Semigroups . . . . . . . . . . . . . . . . . . . . . . . 3 Jerome A. Goldstein and Rainer Nagel On Nonlocal Perturbations of Integral Kernels. . . . . . . . . . . . . . . . . . 27 Krzysztof Bogdan and Sebastian Sydor Convolution Operators as Generators of One-Parameter Semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Jan Kisyński One-Parameter Semigroups in the Algebra of Slowly Increasing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Jan Kisyński Some Remarks on the Krein-von Neumann Extension of Different Laplacians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Delio Mugnolo On Strong Convergence to Ergodic Projection for Perturbed Substochastic Semigroups. . . . . . . . . . . . . . . . . . . . . . . 89 Mustapha Mokhtar-Kharroubi On Robustness of Strongly Stable Semigroups with Spectrum on iR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Lassi Paunonen xi