Geometric means of quasi-Toeplitz matrices

Abstract

We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices $A=(a_{i,j})_{i,j=1,2,\ldots }$ 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 $\sum _{k=-\infty }^{\infty } a_k e^{{{\mathfrak {i}}}k t}$, in the sense that $(T(a))_{i,j}=a_{j-i}$. If a is real valued and essentially bounded, then these matrices represent bounded self-adjoint operators on $\ell ^2$. We prove that if $a_1,\ldots ,a_p$ are continuous and positive functions, or are in the Wiener algebra with some further conditions, then matrix geometric means, such as the ALM, the NBMP and the weighted mean of quasi-Toeplitz positive definite matrices associated with $a_1,\ldots ,a_p$, are quasi-Toeplitz matrices associated with the function $(a_1\cdots a_p)^{\frac{1}{p}}$, which differ only by the compact correction. We introduce numerical algorithms for their computation and show by numerical tests that these operator means can be effectively approximated numerically.