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Variational principles for symmetric bilinear forms
Author(s) -
Danciger Jeffrey,
Garcia Stephan Ramon,
Putinar Mihai
Publication year - 2008
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510641
Subject(s) - mathematics , hilbert space , eigenvalues and eigenvectors , spectrum (functional analysis) , toeplitz matrix , bilinear interpolation , operator (biology) , minimax , pure mathematics , domain (mathematical analysis) , bilinear form , sequence (biology) , space (punctuation) , planar , mathematical analysis , mathematical optimization , quantum mechanics , computer science , biochemistry , statistics , physics , chemistry , genetics , repressor , biology , transcription factor , gene , computer graphics (images) , operating system
Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant's minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general framework to the Friedrichs operator of a planar domain and to Toeplitz operators and their compressions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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