On linear continuous operators between distinguished spaces Cp(X)

Jerzy Ka̧kol, Arkady Leiderman

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, a locally convex space Cp(X) is distinguished if and only if X is a Δ -space. If there exists a linear continuous surjective mapping T: Cp(X) → Cp(Y) and Cp(X) is distinguished, then Cp(Y) also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator T: Cp(X) → Cp(Y) above is open? Secondly, we devote a special attention to concrete distinguished spaces Cp([1 , α]) , where α is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping T: Cp([1 , α]) → Cp(Y) is given. We also observe that for every countable ordinal α all closed linear subspaces of Cp([1 , α]) are distinguished, thereby answering an open question posed in Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of Δ -spaces we prove that a linear continuous surjection T: Cp(X) → Ck(X) w, where Ck(X) w denotes the Banach space C(X) endowed with its weak topology, does not exist for every infinite metrizable compact C-space X (in particular, for every infinite compact X⊂ Rn).

Original languageEnglish
Article number199
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Volume115
Issue number4
DOIs
StatePublished - 1 Oct 2021

Keywords

  • Countable ordinal
  • Distinguished locally convex space
  • Linear continuous operator
  • Δ -space

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Computational Mathematics
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'On linear continuous operators between distinguished spaces Cp(X)'. Together they form a unique fingerprint.

Cite this