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Bruno de Malafosse Eberhard Malkowsky Vladimir Rakočević Operators Between Sequence Spaces and Applications Operators Between Sequence Spaces and Applications Bruno de Malafosse Eberhard Malkowsky (cid:129) (cid:129) č ć Vladimir Rako evi Operators Between Sequence Spaces and Applications 123 BrunodeMalafosse Eberhard Malkowsky University of LeHavre (LMAH) Faculty of Management Ore,France Univerzitet Union NikolaTesla Beograd,Serbia Vladimir Rakočević Department ofMathematics University of Niš Niš, Serbia ISBN978-981-15-9741-1 ISBN978-981-15-9742-8 (eBook) https://doi.org/10.1007/978-981-15-9742-8 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this 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, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface The study of operators between sequence spaces is a wide field in modern summability. In general, summability theory deals with a generalization of the concept of convergence of sequences of complex numbers. One of the original ideas was to assign, in some way, a limit to divergent series, by considering a transform, in many cases defined by the use of matrices, rather than the original series.Acentralproblemofinterestisthecharacterizationofclassesofalloperators between sequence spaces: The first result in this area was the famous Toeplitz theorem which established necessary and sufficient conditions on the entries of a matrixtotransformtopreserveconvergence.Theoriginalproofusedtheanalytical method of the gliding hump. The introductionon a large scale offunctional analytic methodsto summability inthe1940sandthedevelopmentoftheFK andBK spacetheorymadethestudyof matrix transformations and linear operators between sequence spaces a rapidly expanding field of interest in summability. More recently, in particular after 2000, the theory of measures on noncompactness was applied in the characterization of compact operators between BK spaces. This book presents recent studies on bounded and compact operators, their underlyingtheoriesandapplicationstoproblemsofthesolutionofinfinitesystems of linear equations in various sequence spaces. It consists of two parts. In the first part, Chaps. 1–3, it presents the modern methods in functional analysis and operator theory and their applications in recent research in the rep- resentationsandcharacterizationsofboundedandcompactlinearoperatorsbetween BK spaces and between matrix domains of triangles and row finite matrices in BK spaces.Inthesecondpart,Chaps.4–7,wepresentapplicationsandresearchresults using the results of the first three chapters. This involves the study of whether infinite matrices A¼ða Þ1 are injective, surjective or bijective, when consid- nk n;k¼1 ered as operators between certain sequence spaces. Thisbookisunique,sinceitconnectsandpresentsthetopicsofthetwopartsin one volume for the first time, to the best of the authors’ knowledge. v vi Preface This book contains relevant parts of several of the authors’ related lectures on graduate and postgraduate levels at universities in Australia, France, Germany, India,Jordan,Mexico,Serbia,Turkey,theUSAandSouthAfrica.Agreatnumber of illustrating examples and remarks were added concerning the presented topics. Thebookcouldalsobeusedasatextbookforgraduateandpostgraduatecourses as a basis for and overview of research in the fields mentioned and is intended to address students, teachers and researchers alike. Chapter 1 is mostly introductory and recalls the basic concepts from functional analysisneededinthebook,suchaslinearmetricandparanormedspaces,FK,BK, AK and AD spaces, multiplier spaces, matrix transformations, measures of noncompactness, in particular, the Hausdorff and Kuratowski measures of noncompactness, and measures of noncompactness of operators between Banach spaces. Although most of the material is standard, almost all proofs are presented of the vital results. InChap.2,westudysequencespacesthathaverecentlybeenintroducedbythe use of infinite matrices. They can be considered as the matrix domains in certain sequence spaces and can be used to define almost all the classical methods of summability as special cases. We apply the results and methods of Chap. 1 to determine their topological properties, bases and various duals; in particular, we establishageneralresultforthedeterminationoftheb-dualofarbitrarytrianglesin arbitrary FK spaces. InChap. 3,wecharacterize matrix transformations on thespaces ofgeneralized weightedmeansandonmatrixdomainsoftrianglesinBK spaces.Wealsoestablish estimates or identities for the Hausdorff measure of noncompactness of matrix transformations from arbitrary BK spaces with AK into c, c and ‘ , and also from 0 1 the matrix domains of an arbitrary triangle in ‘ , c and c into c, c and ‘ . p 0 0 1 Furthermore,wedeterminetheclassesofcompactoperatorsbetweenthespacesjust mentioned. Finally, we establish the representations of the general bounded linear operators from c into itself and from the space bvþof sequences of bounded variationintoc,andthedeterminationoftheclassesofcompactoperatorsbetween them. In Chaps. 5 and 6, we obtain new results on sequence spaces inclusion and on sequence spaces equations using the results of the first part on matrix transformations. This study leads to a lot of results published until now, for instance, in the statisticalconvergenceandinthespectraltheorywheretheoperatorsaredefinedby infinite matrices. We also deal with the solvability of infinite systems of linear equations in various sequence spaces. Here, we use the classical sequence spaces andthegeneralizedCesàroanddifferenceoperatorstoobtainmanycalculationsand simplifications of complicated spaces involving these operators. We also consider the sum and the product of some linear spaces of sequences involving the sets of sequences that are “strongly bounded and summable to zero”. Finally, we obtain new results on “statistical convergence”. Preface vii InChap.7,weconsideraBanachalgebrainwhichwemayobtaintheinverseof an infinite matrix and obtain a new method to calculate the “Floquet exponent”. Furthermore, we determine the solutions of the infinite linear system associated with the Hill equation with a second member and give a method to approximate them. Finally, we present a study of the Mathieu equation which can be written as an infinite tridiagonal linear system of equations. Le Havre, France Bruno de Malafosse Niš, Serbia Eberhard Malkowsky Niš, Serbia Vladimir Rakočević August 2020 Contents 1 Matrix Transformations and Measures of Noncompactness . . . . . . . 1 1.1 Linear Metric and Paranormed Spaces . . . . . . . . . . . . . . . . . . . . 2 1.2 FK and BK Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Matrix Transformations into the Classical Sequence Spaces . . . . 15 1.4 Multipliers and Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Matrix Transformations Between the Classical Sequence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 Crone’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.7 Remarks on Measures of Noncompactness. . . . . . . . . . . . . . . . . 34 1.8 The Axioms of Measures of Noncompactness . . . . . . . . . . . . . . 35 1.9 The Kuratowski and Hausdorff Measures of Noncompactness . . . 37 1.10 Measures of Noncompactness of Operators . . . . . . . . . . . . . . . . 42 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Russian References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2 Matrix Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 Bases of Matrix Domains of Triangles. . . . . . . . . . . . . . . . . . . . 55 2.3 The Multiplier Space MðXR;YÞ . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.4 The a-, b- and c-duals of XR . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.5 The a- and b-duals of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 DðmÞ 2.6 The b-duals of Matrix Domains of Triangles in FK spaces . . . . . 90 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3 Operators Between Matrix Domains. . . . . . . . . . . . . . . . . . . . . . . . . 105 3.1 Matrix Transformations on Wðu;v;XÞ . . . . . . . . . . . . . . . . . . . . 105 3.2 Matrix Transformations on XT . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.3 Compact Matrix Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.4 The Class KðcÞ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 ix x Contents 3.5 Compact Operators on the Space bvþ . . . . . . . . . . . . . . . . . . . . 145 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4 Computations in Sequence Spaces and Applications to Statistical Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.1 On Strong s-Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.2 Sum and Product of Spaces of the Form sn, s0n, or sðncÞ . . . . . . . . 162 4.3 Properties of the Sequence CðsÞs. . . . . . . . . . . . . . . . . . . . . . . . 166 4.4 Some Properties of the Sets ssðDÞ, s0sðDÞ and sðscÞðDÞ . . . . . . . . . 172 4.5 The Spaces wsðkÞ, w(cid:2)sðkÞ and w(cid:3)sðkÞ. . . . . . . . . . . . . . . . . . . . . . 174 4.6 Matrix Transformations From wsðkÞþwmðlÞ into sc . . . . . . . . . . 179 4.7 On the Sets csðk;lÞ, c(cid:2)sðk;lÞ and c(cid:3)sðk;lÞ . . . . . . . . . . . . . . . . . 180 4.8 Sets of Sequences of the Form ½A ;A (cid:4) . . . . . . . . . . . . . . . . . . . 183 1 2 4.9 Extension of the Previous Results . . . . . . . . . . . . . . . . . . . . . . . 189 4.10 Sets of Sequences that are Strongly s-Bounded With Index p . . . 192 4.11 Computations in Ws and Ws0 and Applications to Statistical Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 4.12 Calculations in New Sequence Spaces . . . . . . . . . . . . . . . . . . . . 204 4.13 Application to A(cid:5)Statistical Convergence . . . . . . . . . . . . . . . . . 207 4.14 Tauberian Theorems for Weighted Means Operators. . . . . . . . . . 214 4.15 The Operator CðkÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 5 Sequence Spaces Inclusion Equations . . . . . . . . . . . . . . . . . . . . . . . . 229 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5.2 The (SSIE) F (cid:6)EaþFx0 with e2F and F0 (cid:6)MðF;F0Þ . . . . . . . 236 5.3 The (SSIE) F (cid:6)EaþFx0 with E;F;F0 2fc0;c;s1;‘p;w0;w1g. . . 238 5.4 Some (SSIE) and (SSE) with Operators . . . . . . . . . . . . . . . . . . . 245 5.5 The (SSIE) F (cid:6)EaþFx0 for e62F. . . . . . . . . . . . . . . . . . . . . . . 250 5.6 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 6 Sequence Space Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 6.2 The (SSE) E þF ¼F with e2F . . . . . . . . . . . . . . . . . . . . . 271 a x b 6.3 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 6.4 The (SSE) with Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 6.5 Some (SSE’s) with the Operators D and R. . . . . . . . . . . . . . . . . 289 6.6 The Multiplier MððEaÞD;FÞ and the (SSIE) Fb (cid:6)ðEaÞDþFx. . . . 297 6.7 The (SSE) ðEaÞDþsxðcÞ ¼sbðcÞ. . . . . . . . . . . . . . . . . . . . . . . . . . . 300 6.8 More Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Contents xi 7 Solvability of Infinite Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 315 7.1 Banach Algebras of Infinite Matrices. . . . . . . . . . . . . . . . . . . . . 315 7.2 Solvability of the Equation Ax¼b . . . . . . . . . . . . . . . . . . . . . . 320 7.3 Spectra of Operators Represented by Infinite Matrices . . . . . . . . 326 7.4 Matrix Transformations in vðDmÞ. . . . . . . . . . . . . . . . . . . . . . . . 330 7.5 The Equation Ax¼b, Where A Is a Tridiagonal Matrix . . . . . . . 338 7.6 Infinite Linear Systems with Infinitely Many Solutions . . . . . . . . 343 7.7 The Hill and Mathieu Equations . . . . . . . . . . . . . . . . . . . . . . . . 349 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Appendix: Inequalities .. .... ..... .... .... .... .... .... ..... .... 359 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 363