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Automatic boundedness of some operators between ordered and topological vector spaces

Eduard Emelyanov

TL;DR

The paper studies the boundedness and continuity of order-to-topology operators between ordered vector spaces and topological vector spaces, and develops collective convergence notions to unify their behavior. It establishes key inclusions and identities among operator classes, notably that order-to-topology bounded operators are ru-to-topology continuous and coincide with ru-to-topology continuous operators when the positive cone is generating. It proves automatic boundedness and a Uniform Boundedness Principle for collectively bounded operator families, and connects order-to-topology boundedness with norm-boundedness in certain cone settings. These results extend prior work to broader TVS contexts, yield structural equalities under cone hypotheses, and provide tools for analyzing positivity-preserving operator families and semigroups in ordered spaces.

Abstract

We study topological boundedness of order-to-topology bounded and order-to-topology continuous operators from ordered vector spaces to topological vector spaces. The uniform boundedness principle for such operators is investigated.

Automatic boundedness of some operators between ordered and topological vector spaces

TL;DR

The paper studies the boundedness and continuity of order-to-topology operators between ordered vector spaces and topological vector spaces, and develops collective convergence notions to unify their behavior. It establishes key inclusions and identities among operator classes, notably that order-to-topology bounded operators are ru-to-topology continuous and coincide with ru-to-topology continuous operators when the positive cone is generating. It proves automatic boundedness and a Uniform Boundedness Principle for collectively bounded operator families, and connects order-to-topology boundedness with norm-boundedness in certain cone settings. These results extend prior work to broader TVS contexts, yield structural equalities under cone hypotheses, and provide tools for analyzing positivity-preserving operator families and semigroups in ordered spaces.

Abstract

We study topological boundedness of order-to-topology bounded and order-to-topology continuous operators from ordered vector spaces to topological vector spaces. The uniform boundedness principle for such operators is investigated.
Paper Structure (2 sections, 17 theorems)

This paper contains 2 sections, 17 theorems.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Theorem 2.1

Let $X$ be an OVS and $(Y,\tau)$ a TVS. Then $\text{\bf L}_{o{\tau}b}(X,Y)\subseteq\text{\bf L}_{r{\tau}c}(X,Y)$. If additionally $X_+$ is generating then $\text{\bf L}_{o{\tau}b}(X,Y)=\text{\bf L}_{r{\tau}c}(X,Y)$.

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • ...and 13 more