LEADER 04421nam 2200505 450 001 996499867103316 005 20230512095424.0 010 $a3-031-19436-5 035 $a(MiAaPQ)EBC7150620 035 $a(Au-PeEL)EBL7150620 035 $a(CKB)25510405700041 035 $a(OCoLC)1352972743 035 $a(PPN)26634822X 035 $a(EXLCZ)9925510405700041 100 $a20230416d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aIndex theory beyond the Fredholm case /$fAlan L. Carey, Galina Levitina 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (186 pages) 225 1 $aLecture notes in mathematics ;$vVolume 2323 311 08$aPrint version: Carey, Alan Index Theory Beyond the Fredholm Case Cham : Springer,c2023 9783031194351 320 $aIncludes bibliographical references. 327 $aIntro -- Preface -- Acknowledgements -- Notations -- Contents -- 1 Introduction -- 1.1 Motivation and Background -- 1.2 An Overview of Recent Results -- 1.3 Discussion of the Methods and the Applications in These Notes -- 1.4 Summary of the Exposition -- 2 Double Operator Integrals -- 2.1 Double Operator Integrals in the Discrete Setting -- 2.2 Double Operator Integrals in the General Setting -- 2.3 Double Operator Integrals for Resolvent Comparable Operators -- 2.4 Continuity of Double Operator Integrals with Respect to the Operator Parameters -- 3 The Model Operator and Its Approximants -- 3.1 The Class of p-Relative Trace-Class Perturbations -- 3.2 Main Setting and Assumptions -- 4 The Spectral Shift Function -- 4.1 An Introduction to the Theory of the Spectral Shift Function -- 4.1.1 Perturbation Determinants -- 4.1.2 M. G. Krein' s Construction of the Spectral Shift Function -- 4.1.3 Properties of the Spectral Shift Function -- 4.2 More General Classes of Perturbations -- 4.2.1 Spectral Shift Function for Unitary Operators -- 4.2.2 Spectral Shift Function for Resolvent Comparable Operators -- 4.2.3 Invariance Principle -- 4.2.4 Spectral Shift Function for m-Resolvent Comparable Operators -- 4.3 Continuity of the Spectral Shift Function with Respect to the Operator Parameter -- 4.4 Representation of the Spectral Shift Function via a Regularised Perturbation Determinant -- 4.5 Spectral Shift Functions for the Pairs (A+,A-), (H2,H1) -- 5 Spectral Flow -- 5.1 Phillips' Definition of Spectral Flow and Analytic Formulas -- 5.1.1 The Variation of eta Formula -- 5.1.2 A Review of Analytic Formulas for Spectral Flow -- 5.2 The Relation Between the Spectral Shift Function and the Spectral Flow -- 5.3 Generalised Spectral Flow -- 6 The Principal Trace Formula and Its Applications -- 6.1 A Brief History of the Principal Trace Formula. 327 $a6.2 Proving the Principal Trace Formula -- 6.3 A Generalised Pushnitski Formula -- 6.4 The Witten Index -- 6.4.1 Preliminaries -- 6.4.2 The Formula in Terms of the Spectral Shift Function -- 6.5 Cyclic Homology and Invariance -- 6.5.1 How the Witten Index Relates to This -- 6.5.2 Higher Schatten Classes -- 6.6 The Anomaly in Terms of the Spectral Shift Function -- 6.6.1 The Origin of the Notion of an `Anomaly' -- 6.6.2 Relationship to the Spectral Shift Function -- 7 Examples -- 7.1 The Dirac Operator in Rd -- 7.1.1 The Setting -- 7.1.2 Verification of Hypothesis 3.2.5 -- 7.1.3 The Index of DA -- 7.1.4 Behaviour of the Spectral Shift Function for the Massless Dirac Operator -- 7.2 A Compact One-Dimensional Example -- 7.2.1 The Setting -- 7.2.2 Spectral Shift Function of the Pair (D?+?, D?) -- 7.2.3 The Index of the Operator DA -- 7.2.4 Spectral Flow Along the path {D?+?(t)?}tR -- 7.2.5 The Anomaly for the Operator DA -- References. 410 0$aLecture notes in mathematics (Springer-Verlag) ;$vVolume 2323. 606 $aIndex theory (Mathematics) 606 $aTeoria de l'índex (Matemàtica)$2thub 608 $aLlibres electrònics$2thub 615 0$aIndex theory (Mathematics) 615 7$aTeoria de l'índex (Matemàtica) 676 $a512.556 700 $aCarey$b Alan L.$061721 702 $aLevitina$b Galina 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996499867103316 996 $aIndex theory beyond the Fredholm case$93088867 997 $aUNISA