Bushell's equations and polar decompositions

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.