On linear continuous operators between distinguished spaces Cp(X)

0546784 - MÚ 2022 RIV ES eng J - Journal Article
Kąkol, Jerzy - Leiderman, A. G.
On linear continuous operators between distinguished spaces Cp(X).
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Roč. 115, č. 4 (2021), č. článku 199. ISSN 1578-7303. E-ISSN 1579-1505
R&D Projects: GA ČR(CZ) GF20-22230L
Institutional support: RVO:67985840
Keywords : countable ordinal * distinguished locally convex space * linear continuous operator
OECD category: Pure mathematics
Impact factor: 2.276, year: 2021
Method of publishing: Limited access
https://doi.org/10.1007/s13398-021-01121-4

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).
Permanent Link: http://hdl.handle.net/11104/0323163