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Bushell's equations and polar decompositions
Author(s) -
Lim Yongdo
Publication year - 2009
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.200610820
Subject(s) - mathematics , positive definite matrix , polar decomposition , invertible matrix , pure mathematics , homeomorphism (graph theory) , hilbert space , banach space , operator (biology) , unitary state , sesquilinear form , mathematical analysis , combinatorics , polar , hermitian matrix , eigenvalues and eigenvectors , biochemistry , physics , chemistry , repressor , quantum mechanics , astronomy , transcription factor , gene , political science , law
We show that for any real number t with t ≠ ±1, every invertible operator M on a Hilbert space admits a new polar decomposition M = PUP –t where P is positive definite and U is unitary, and that the corresponding polar map is homeomorphism. The positive definite factor P of M appears as the negative square root of the unique positive definite solution of the nonlinear operator equation X t = M * XM . This extends the classical matrix and operator polar decomposition when t = 0. For t = ± 1, it is shown that the positive definite solution sets of X ±1 = M * XM form geodesic submanifolds of the Banach–Finsler manifold of positive definite operators and coincide with fixed point sets of certain non‐expansive mappings, respectively (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)