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A Generalization of Lévy's Theorem on Positive Matrix Semigroups

Moritz Gerlach

Abstract

We generalize a fundamental theorem on positive matrix semigroups stating that each component is either strictly positive for all times or identically zero ("Lévy's Theorem"). Our proof of this fact that does not require the matrices to be Markovian nor to be continuous at time zero. We also provide a formulation of this theorem in the terminology of one-parameter operator semigroups on sequence spaces.

A Generalization of Lévy's Theorem on Positive Matrix Semigroups

Abstract

We generalize a fundamental theorem on positive matrix semigroups stating that each component is either strictly positive for all times or identically zero ("Lévy's Theorem"). Our proof of this fact that does not require the matrices to be Markovian nor to be continuous at time zero. We also provide a formulation of this theorem in the terminology of one-parameter operator semigroups on sequence spaces.
Paper Structure (4 sections, 12 theorems, 33 equations)

This paper contains 4 sections, 12 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Positive Matrix Semigroups
  3. Operator Semigroups on Sequence Spaces
  4. Analytic Semigroups

Key Result

Theorem 1.1

Let $p_{ij}$ be a positive matrix semigroup over a countable index set $I$. Then for all $i,j \in I$

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 13 more