On the boundedness of infinite matrix products with alternating factors from two sets of matrices

TL;DR

This paper investigates conditions under which infinite products of matrices with alternating factors from two sets remain bounded, exploring when such boundedness can be uniformly guaranteed across all sequences.

Contribution

It provides conditions and scenarios where the norms of these matrix products are uniformly bounded, advancing understanding of matrix product boundedness with alternating factors.

Findings

Identifies cases where matrix product norms are uniformly bounded.

Shows that boundedness depends on properties of the matrix sets.

Highlights open problems in the general case of boundedness.

Abstract

We consider the question of the boundedness of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices, $A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to an appropriate choice of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices $\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ for which the sequence of matrix products $\{A_{n}B_{n}\cdots A_{1}B_{1}\}_{n=1}^{\infty}$ is norm bounded. Some situations are described in which in this case the norms of matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ are uniformly bounded, that is, $\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le C$ for all natural numbers $n$, where $C>0$ is some constant independent of the sequence $\{A_{i}\}$ and the corresponding sequence $\{B_{i}\}$. In the general case, the question of the validity of the corresponding statement remains open.