  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.
