A Birkhoff contraction formula with applications to Riccati Equations

The positive symplectic operators on a Hilbert space E oplus E give rise to linear fractional transformations on the open convex cone of positive definite operators on E. These fractional transformations contract a natural Finsler metric, the Thompson or part metric, on the convex cone. More precisely, the constants of contraction for these positive fractional operators satisfy the classical Birkhoff formula: the Lipschitz constant for the corresponding linear fractional transformations on the cone of positive definite operators is equal to the hyperbolic tangent of one fourth the diameter of the image. By means of the close connections between sympletic operators and Riccati equations, this result and the associated machinery can be readily applied to obtain convergence results and rates for discrete algebraic Riccati equations and Riccati differential equations.