Bushell's equations and polar decompositions

Abstract

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 Xt = 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)