DeepAI

# Geometric means of quasi-Toeplitz matrices

We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices A=(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 ∑_ℓ=-∞^∞ a_ℓ e^𝔦ℓ t, in the sense that (T(a))_i,j=a_j-i. If a is and essentially bounded, then these matrices represent 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 NBMP and 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.

• 4 publications
• 2 publications
• 7 publications
03/12/2022

### Computing eigenvalues of semi-infinite quasi-Toeplitz matrices

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the form A=T(a...
07/05/2019

### Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations

We consider a class of linear matrix equations involving semi-infinite m...
04/07/2019

### Statistical Meaning of Mean Functions

The basic properties of the Fisher information allow to reveal the stati...
09/20/2022

### Quasi-Perron-Frobenius property of a class of saddle point matrices

The saddle point matrices arising from many scientific computing fields ...
03/08/2022

### Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality

We introduce a formulation of quantum theory (QT) as a general probabili...
03/04/2019

### Krylov Iterative Methods for the Geometric Mean of Two Matrices Times a Vector

In this work, we are presenting an efficient way to compute the geometri...
09/25/2019

### A computational framework for two-dimensional random walks with restarts

The treatment of two-dimensional random walks in the quarter plane leads...