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Automatic time continuity of positive matrix and operator semigroups

Jochen Glück

TL;DR

This work addresses automatic continuity of semigroups without any measurability assumptions. It proves a finite-dimensional result: a positive matrix semigroup with boundedness near zero is necessarily continuous, and extends this to positive operators on sequence spaces. The key techniques combine Perron-Frobenius spectral arguments, universal-net convergence, and the Jacobs–de Leeuw–Glicksberg decomposition to exclude nontrivial limits and force convergence to the identity as time approaches zero. Collectively, the results remove measurability as a prerequisite for automatic continuity in these settings and connect to $C_0$-semigroup theory and Markov-embedding-type questions.

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.

Automatic time continuity of positive matrix and operator semigroups

TL;DR

This work addresses automatic continuity of semigroups without any measurability assumptions. It proves a finite-dimensional result: a positive matrix semigroup with boundedness near zero is necessarily continuous, and extends this to positive operators on sequence spaces. The key techniques combine Perron-Frobenius spectral arguments, universal-net convergence, and the Jacobs–de Leeuw–Glicksberg decomposition to exclude nontrivial limits and force convergence to the identity as time approaches zero. Collectively, the results remove measurability as a prerequisite for automatic continuity in these settings and connect to -semigroup theory and Markov-embedding-type questions.

Abstract

We consider a matrix semigroup without assuming any measurability properties and show that, if is bounded close to and entrywise for all , then is continuous. This complements classical results for the scalar-valued case. We also prove an analogous result if takes values in the positive operators over a sequence space.

Paper Structure

This paper contains 3 sections, 5 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Automatic continuity of matrix semigroups
  3. Automatic continuity on sequence spaces

Key Result

Proposition 1.1

Let $T: [0,\infty) \to \mathbb{R}$ satisfy the functional equation eq:fe-scalar and assume that $T(t) \not= 0$ for at least one $t \in (0,\infty)$. If $\sup_{t \in [0,1]} \left\lvert T(t) \right\rvert < \infty$, then $T$ is continuous (and hence $T(t) = e^{ta}$ for a number $a \in \mathbb{R}$ and al

Theorems & Definitions (12)

  • Proposition 1.1: Continuity of scalar semigroups
  • Example 1.2
  • proof
  • proof : Proof of Proposition \ref{['prop:scalar']}
  • Theorem 2.1: Continuity of positive matrix semigroups
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:matrix-pos']}
  • Theorem 3.1: Automatic continuity on sequence spaces
  • proof
  • ...and 2 more