A monotone geometric mean for a class of Toeplitz matrices

In this paper, using Laurent operators and Fourier coefficients of their symbol functions, we introduce a geometric mean for a large class of n×n positive semi-definite Toeplitz matrices which satisfies the monotonicity property. The cost of our approach in term of arithmetic operations for m matrices is of the order O(mn2). This definition preserves the structure, is simple to calculate, preserves monotonicity and satisfies some other Ando-Li-Mathias properties.