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 $\sum_{\ell=-\infty}^{\infty} a_\ell e^{\mathfrak i \ell t}$, in the sense that $(T(a))_{i,j}=a_{j-i}$. If $a$ is \rv\ and essentially bounded, then these matrices represent bounded self-adjoint operators on $\ell^2$. 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 $a_1,\ldots,a_p$ 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 NBMP and the Karcher mean of quasi-Toeplitz positive definite matrices associated with $a_1,\ldots,a_p$, are quasi-Toeplitz matrices associated with the geometric mean $(a_1\cdots a_p)^{1/p}$, which differ only by the compact correction. We show by numerical tests that these operator means can be practically approximated.

翻译：我们研究的是半非硬基质基质的几何类型准托普利茨基质,即半非硬基质基质 $A=(a)i,j=1,j=2,hldots}美元(美元=T(a)+E$,美元=T(a)+E$,美元代表一个小型操作员),而$T(a)美元是半非硬基质的托普利茨基质,与函数美元相关联的半非固定基质。我们研究的是,美元是一个连续的函数, 纯基质基质与典型的托普利茨基质相吻合, 基质与基质基质的基质(a)x(a), 基质的基质和基质的基质的基质是正值, 基质的基质的基质和基质的基质的基质、 基质的基质的基质的基质和基质的基质的基质的基质。