On algebras generated by positive operators

TL;DR

This paper investigates the structure and properties of algebras generated by positive matrices, extending recent theorems, providing transparent proofs, and exploring algebra dimensions and triangularizability conditions.

Contribution

It offers new proofs and extensions of existing theorems on positive matrix algebras, including explicit descriptions of generated algebra dimensions and conditions for triangularizability.

Findings

The algebra generated by certain positive idempotent matrices has dimension at most 9.

Examples show generated algebra dimensions can be 2n or 2n-1 depending on n.

The algebra generated by multiple positive matrices is triangularizable under specific conditions.

Abstract

We study algebras generated by positive matrices, i.e., matrices with nonnegative entries. Some of our results hold in more general setting of vector lattices. We reprove and extend some theorems that have been recently shown by Kandi\'{c} and \v{S}ivic. In particular, we give a more transparent proof of their result that the unital algebra generated by positive idempotent matrices $E$ and $F$ such that $E F \ge F E$ is equal to the linear span of the set $\{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\}$, and so its dimension is at most $9$. We give examples of two positive idempotent matrices that generate unital algebra of dimension $2n$ if $n$ is even, and of dimension $(2n - 1)$ if $n$ is odd. We also prove that the algebra generated by positive matrices $B_1$, $B_2$, $\ldots$, $B_k$ is triangularizable if $A B_i \ge B_i A$ ($i=1,2, \ldots, k$) for some positive matrix $A$ with distinct eigenvalues.