A geometric mean for Toeplitz and Toeplitz-block block-Toeplitz matrices

Using the symbol functions and their associated Fourier series, we introduce a new definition of geometric mean for all positive semi-definite Toeplitz matrices and positive semi-definite block-Toeplitz matrices with Toeplitz structured blocks (TBBT). The proposed definition preserves the structure of the matrices and satisfies some important Ando-Li-Mathias properties, such as monotonicity and continuity. Also, it has low cost and is simple to calculate.