Geometric means of quasi-Toeplitz matrices – Jie Meng (University of Pisa)


We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matricesA = (a_{i,j}) i,j=1,2,… of the form A = T(a) + E, where E represents a compact operator, and T(a) is a semi-infinite Toeplitz matrix associated with the function a, with Fourier series \sum_{l} a_l e^{ilt} , in the sense that (T(a))_{i,j} = a_{j-i}. If a is real valued and essentially bounded, then these matricesrepresent bounded self-adjoint operators on l^2 . We consider the case where a is a continuous function, where quasi-Toeplitz matrices coincide with a classical Toeplitz algebra, and the case where a is in the Wiener algebra, that is, has absolutely convergent Fourier series. We prove that if a_1, … , a_p are continuous and positive functions, or are in the Wiener algebra with some further conditions, then means of geometric type, such as the ALM, the NBMPand the Karcher mean of quasi-Toeplitz positive definite matrices associated with a_1, … , a_p, are quasi-Toeplitz matrices associated with the geometric mean (a_1 … a_p)^{1/p}, which differ only by the compact correction. We show by numerical tests that these operator means can be practically approximated.

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