LEADER 06791nam 22008175 450 001 9910254071903321 005 20251116150418.0 010 $a3-319-27323-X 024 7 $a10.1007/978-3-319-27323-5 035 $a(CKB)3710000000636326 035 $a(EBL)4498655 035 $a(SSID)ssj0001666024 035 $a(PQKBManifestationID)16455087 035 $a(PQKBTitleCode)TC0001666024 035 $a(PQKBWorkID)15000339 035 $a(PQKB)10000989 035 $a(DE-He213)978-3-319-27323-5 035 $a(MiAaPQ)EBC4498655 035 $z(PPN)258862602 035 $a(PPN)193445263 035 $a(EXLCZ)993710000000636326 100 $a20160406d2016 u| 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aNon-Archimedean operator theory /$fby Toka Diagana, François Ramaroson 205 $a1st ed. 2016. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2016. 215 $a1 online resource (163 p.) 225 1 $aSpringerBriefs in Mathematics,$x2191-8198 300 $aDescription based upon print version of record. 311 08$a3-319-27322-1 320 $aIncludes bibliographical references and index. 327 $aPreface; Acknowledgments; Contents; 1 Non-Archimedean Valued Fields; 1.1 Valuation; 1.1.1 Definitions and First Properties; 1.1.2 The Topology Induced by a Valuation on K; 1.1.3 Non-Archimedean Valuations; 1.1.4 Some Analysis on a Complete Non-Archimedean Valued Field; 1.1.5 The Order Function for a Discrete Valuation; 1.2 Examples; 1.2.1 Examples of Archimedean Valuation; 1.2.2 Examples of Non-Archimedean Valued Fields; 1.3 Additional Properties of Non-ArchimedeanValued Fields; 1.4 Some Remarks on Krull Valuations; 1.5 Bibliographical Notes; 2 Non-Archimedean Banach Spaces 327 $a2.1 Non-Archimedean Norms2.2 Non-Archimedean Banach Spaces; 2.3 Free Banach Spaces; 2.4 The p-adic Hilbert Space E?; 2.5 Bibliographical Notes; 3 Bounded Linear Operators in Non-Archimedean Banach Spaces; 3.1 Bounded Linear Operators; 3.1.1 Definitions and Examples; 3.1.2 Basic Properties; 3.1.3 Bounded Linear Operators in Free Banach Spaces; 3.2 Additional Properties of Bounded Linear Operators; 3.2.1 The Inverse Operator; 3.2.2 Perturbations of Orthogonal Bases Using the Inverse Operator; 3.2.3 The Adjoint Operator; 3.3 Finite Rank Linear Operators; 3.3.1 Basic Definitions 327 $a3.3.2 Properties of Finite Rank Operators3.4 Completely Continuous Linear Operators; 3.4.1 Basic Properties; 3.4.2 Completely Continuous Linear Operators on E?; 3.5 Bounded Fredholm Linear Operators; 3.5.1 Definitions and Examples; 3.5.2 Properties of Fredholm Operators; 3.6 Spectral Theory for Bounded Linear Operators; 3.6.1 The Spectrum of a Bounded Linear Operator; 3.6.2 The Essential Spectrum of a Bounded Linear Operator; 3.7 Bibliographical Notes; 4 The Vishik Spectral Theorem; 4.1 The Shnirel'man Integral and Its Properties; 4.1.1 Basic Definitions; 4.1.2 The Shnirel'man Integral 327 $a4.2 Distributions with Compact Support4.3 Cauchy-Stieltjes and Vishik Transforms; 4.4 Analytic Bounded Linear Operators; 4.5 Vishik Spectral Theorem; 4.6 Bibliographical Notes; 5 Spectral Theory for Perturbations of Bounded Diagonal Linear Operators; 5.1 Spectral Theory for Finite Rank Perturbations of Diagonal Operators; 5.1.1 Introduction; 5.1.2 Spectral Analysis for the Class of Operators T = D + K; 5.1.3 Spectral Analysis for the Class of Operators T = D + F; 5.2 Computation of ?e(D); 5.3 Spectrum of T = D + F; 5.4 Examples; 5.5 Bibliographical Notes; 6 Unbounded Linear Operators 327 $a6.1 Unbounded Linear Operators on a Non-archimedean Banach Space6.2 Closed Linear Operators; 6.3 The Spectrum of an Unbounded Operator; 6.4 Unbounded Fredholm Operators; 6.5 Bibliographical Notes; 7 Spectral Theory for Perturbations of Unbounded Linear Operators; 7.1 Introduction; 7.2 Spectral Analysis for the Class of Operators T = D + K; 7.3 Spectral Analysis for the Class of Operators T = D + F; 7.4 Computation of ?e(D); 7.5 Main Result; 7.6 Bibliographical Notes; A The Shnirel'man Integral; A.1 Distributions with Compact Support; A.2 Cauchy-Stieltjes and Vishik Transforms; References 327 $aIndex 330 $aThis book focuses on the theory of linear operators on non-Archimedean Banach spaces. The topics treated in this book range from a basic introduction to non-Archimedean valued fields, free non-Archimedean Banach spaces, bounded and unbounded linear operators in the non-Archimedean setting, to the spectral theory for some classes of linear operators. The theory of Fredholm operators is emphasized and used as an important tool in the study of the spectral theory of non-Archimedean operators. Explicit descriptions of the spectra of some operators are worked out. Moreover, detailed background materials on non-Archimedean valued fields and free non-Archimedean Banach spaces are included for completeness and for reference. The readership of the book is aimed toward graduate and postgraduate students, mathematicians, and non-mathematicians such as physicists and engineers who are interested in non-Archimedean functional analysis. Further, it can be used as an introduction to the study of non-Archimedean operator theory in general and to the study of spectral theory in other special cases. . 410 0$aSpringerBriefs in Mathematics,$x2191-8198 606 $aOperator theory 606 $aFunctional analysis 606 $aAlgebra 606 $aField theory (Physics) 606 $aFunctions of real variables 606 $aOperator Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M12139 606 $aFunctional Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12066 606 $aField Theory and Polynomials$3https://scigraph.springernature.com/ontologies/product-market-codes/M11051 606 $aReal Functions$3https://scigraph.springernature.com/ontologies/product-market-codes/M12171 615 0$aOperator theory. 615 0$aFunctional analysis. 615 0$aAlgebra. 615 0$aField theory (Physics) 615 0$aFunctions of real variables. 615 14$aOperator Theory. 615 24$aFunctional Analysis. 615 24$aField Theory and Polynomials. 615 24$aReal Functions. 676 $a515.724 700 $aDiagana$b Toka$4aut$4http://id.loc.gov/vocabulary/relators/aut$0756010 702 $aRamaroson$b François$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254071903321 996 $aNon-Archimedean Operator Theory$92235914 997 $aUNINA