Automatic time continuity of positive matrix and operator semigroups

TL;DR

This paper proves that positive matrix and operator semigroups, which are bounded near zero, are automatically continuous without assuming measurability, extending classical scalar results to matrix and operator contexts.

Contribution

It establishes the automatic continuity of positive matrix and operator semigroups under boundedness near zero, without requiring measurability assumptions.

Findings

Positive matrix semigroups bounded near zero are continuous.

Analogous continuity result holds for positive operator semigroups on sequence spaces.

Extends classical scalar continuity results to matrix and operator settings.

Abstract

We consider a matrix semigroup $T: [0,\infty) \to \mathbb{R}^{d \times d}$ without assuming any measurability properties and show that, if $T$ is bounded close to $0$ and $T(t) \ge 0$ entrywise for all $t$, then $T$ is continuous. This complements classical results for the scalar-valued case. We also prove an analogous result if $T$ takes values in the positive operators over a sequence space.