TY - JOUR
T1 - On linear continuous operators between distinguished spaces Cp(X)
AU - Ka̧kol, Jerzy
AU - Leiderman, Arkady
N1 - Funding Information:
Jerzy Ka̧kol: supported by the GAČR project 20-22230L and RVO: 67985840. The authors thank J. C. Ferrando for providing a short argument in Remark (iii) and W. B. Johnson and T. Kania for a very helpful advice and discussion about Theorem .
Publisher Copyright:
© 2021, The Author(s) under exclusive licence to The Royal Academy of Sciences, Madrid.
PY - 2021/10/1
Y1 - 2021/10/1
N2 - 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).
AB - 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).
KW - Countable ordinal
KW - Distinguished locally convex space
KW - Linear continuous operator
KW - Δ -space
UR - http://www.scopus.com/inward/record.url?scp=85115337368&partnerID=8YFLogxK
U2 - 10.1007/s13398-021-01121-4
DO - 10.1007/s13398-021-01121-4
M3 - Article
AN - SCOPUS:85115337368
SN - 1578-7303
VL - 115
JO - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
JF - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
IS - 4
M1 - 199
ER -