On algebras generated by positive operators

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 Kandić and Šivic. 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.