Block numerical ranges [Elektronische Ressource] / von Markus Wagenhofer

Block Numerical Rangesvon Markus WagenhoferDissertationzur Erlangung des Grades eines Doktors derNaturwissenschaften– Dr.rer.nat. –Vorgelegt im Fachbereich 3 (Mathematik & Informatik)der Universit¨ at Bremenim Januar 2007Datum des Promotionskolloqiums: 14. Februar 2007Gutachter: Prof.Dr. Christiane Tretter (Universit¨ at Bern)Prof.Dr. Heinz Langer (TU Wien)ContentsIntroduction 31 The block numerical range of bounded operators 91.1 Aboutdecompositions.......................... 91.2 The definition and known results ....................11.3 Continuity properties of W .......................14H1.4 Connected components of W (A)17H1.5 Theblockdeterminantset........................242 The block numerical range of operator functions 272.1 The definition and simple properties ..................282.2 Spectralinclusion.............................312.3 Thenormoftheresolvent343 The block numerical range of operator polynomials 373.1 On the boundedness of W (P) .....................42H3.2 Connectedcomponentsandthenormoftheresolvent.........43.3 Thecompanionpolynomial........................464 Block-diagonalization of operators 494.1 Schur complements ............................504.2 Factorization of Schur complements ...................534.3 InvariantsubspacesandRicatiequations ...............615 Corners of block numerical ranges 655.1 Analyticperturbationsofmatrices....................665.2 Corners of W (A) belonging to W (A).................