Bushell's equations and polar decompositions

Abstract

Key wordsWe show that for any real number t with t = ±1, every invertible operator M on a Hilbert space admits a new polar decomposition M = P UP −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.

Invariant metrics, contractions and nonlinear matrix equations

“…Recently in [34], the author developed the matrix golden mean of positive definite matrices which is defined by A…”

An iterative method to solve the nonlinear matrix equation $X^p =A+\sum\limits_{i = 1}^m {M_i^*(B+X^{-1})^{-1}{M_i}} $

“…stuidied by [3][4][5][6] and the more related matrix equations like X n = M XM * and X n = A + M X −1 M * treated in [7][8][9][10].…”