February 9, 2021

A new preprint is out: "Geometric means of quasi-Toeplitz matrices", with Dario Bini and Jie Meng.
In the paper we show that the most common definitions of matrix geometric mean can be extended to self-adjoint quasi-Toeplitz matrices, that are operators written as a sum of a semi-infinite Toeplitz matrix T(a) and a self-adjoint compact operator. We show also that these infinite-dimensional objects can be effectively approximated and computed using the quasi-Toeplitz technology.
This work has been a little bit hard since I moved on a not so familiar topic (operator theory), but the efforts have been paid with a better knowledge, as usual, but what I like most is that we were able to work out all the difficulties and, apparently, we did not leave incomplete parts. Moreover, the proofs we got were concise, after a long refinement.

One of the most beautiful results is that if T(a) is a semi-infinite Toeplitz matrix associated with the Fourier coefficients of the continuous and real valued function a and if f is a function defined on the range of a, we have that f(T(a))-T(f(a)) is a self-adjoint compact operator.
The most striking fact is that the hypotheses are very mild. A similar result was known for f analytic and for a very regular function a.

Finally, it has been the opportunity to work again with Dario, with whom I have a number of collaboration and to work with Jie, who spent some weeks visiting me in Perugia (before the pandemic). This first ideas have been born in that period and during a visit of Jie and myself to Dario.