The inverse of nonsymmetric two-level Toeplitz operator matrices

The Gohberg–Semencul formula allows one to express the entries of the inverse of a Toeplitz matrix using only a few entries (the first row and the first column) of the inverse matrix, under some nonsingularity condition. In this paper we will provide a two variable generalization of the Gohberg–Semencul formula in the case of a nonsymmetric two-level Toeplitz matrix with a symbol of the form $f(z_1\text{,}{z}_2)=\frac{1}{\overline{P(z_1\text{,}{z}_2)}Q(z_1\text{,}{z}_2)}$ where $P(z_1\text{,}{z}_2)$ and $Q(z_1\text{,}{z}_2)$ are stable polynomials of two variables. We also consider the case of operator valued two-level Toeplitz matrices. In addition, we propose an equation solver involving two-level Toeplitz matrices. Numerical results are included.     

The zero-one law for a complete orthonormal system

A complete orthonormal system of functions $\Theta ={\left\{{\theta }_n\right\}}_{n=1}^{\infty }\text{,}{\theta }_n\in L_{[0\text{,}1]}^{\infty }$ is constructed such that ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ converges almost everywhere on $[0\text{,}1]$ if ${\left\{a_n\right\}}_{n=1}^{\infty }\in l^2$ and ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ diverges a.e. for any ${\left\{a_n\right\}}_{n=1}^{\infty }?l^2$. We also show that for any complete ONS ${\left\{f_n\right\}}_{n=1}^{\infty }$ of functions defined on $[0\text{,}1]$ there exists a fixed non decreasing subsequence ${\left\{n_k\right\}}_{k=1}^{\infty }$ of natural numbers such that for any $f\in L_{[0\text{,}1]}^0$ and some sequence of coefficients ${\left\{b_n\right\}}_{n=1}^{\infty }$,

$\sum _{n=1}^{n_k}{b}_n{f}_n\to f\text{a.e. when}k\to \infty \text{.}$

To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

The finite-step realizability of the joint spectral radius of a pair of d×d matrices one of which being rank-one

We study the finite-step realizability of the joint/generalized spectral radius of a pair of real square matrices ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$, one of which has rank 1, where $2?d<+\infty$. Let $\rho (A)$ denote the spectral radius of a square matrix A. Then we prove that there always exists a finite-length word $(i_1^1\text{,}\dots \text{,}{i}_m^1)\in \{1\text{,}2{\}}^m$, for some finite $m?1$, such that$\sqrt[m]{\rho \left({\mathrm{S}}_{i_1^1}?{\mathrm{S}}_{i_m^1}\right)}={\sup }_{n?1}\left\{{\displaystyle \underset{(i_1\text{,}\dots \text{,}{i}_n)\in \{1\text{,}2{\}}^n}{\max }}\sqrt[n]{\rho ({\mathrm{S}}_{i_1}?{\mathrm{S}}_{i_n})}\right\}\text{.}$In other words, there holds the spectral finiteness property for $\{{\mathrm{S}}_1\text{,}{\mathrm{S}}_2\}$. Explicit formula for computation of the joint spectral radius is derived. This implies that the stability of the switched system induced by $\{{\mathrm{S}}_1\text{,}{\mathrm{S}}_2\}$ is algorithmically decidable in this case.     

A satellite of the grand Furuta inequality and its application

The grand Furuta inequality has the following satellite (SGF;$t\in [0\text{,}1]$), given as a mean theoretic expression:$A?B>0\text{,}t\in [0\text{,}1]?A^{-r+t}{\#}_{\frac{1-t+r}{(p-t)s+r}}(A^t{?}_s{B}^p)?B\text{for}r?t\text{;}p\text{,}s?1\text{,}$where ${\#}_{\alpha }$ is the $\alpha$-geometric mean and ${?}_s$ ($s?[0\text{,}1]$) is a formal extension of ${\#}_{\alpha }$. It is shown that (SGF; $t\in [0\text{,}1]$) has the Löwner–Heinz property, i.e. (SGF; $t=1$) implies (SGF;t) for every $t\in [0\text{,}1]$. Furthermore, we show that a recent further extension of (GFI) by Furuta himself has also the Löwner–Heinz property.