On linear continuous operators between distinguished spaces C_p(X)

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, a locally convex space C_p(X) is distinguished if and only if X is a Δ -space. If there exists a linear continuous surjective mapping T: C_p(X) → C_p(Y) and C_p(X) is distinguished, then C_p(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: C_p(X) → C_p(Y) above is open? Secondly, we devote a special attention to concrete distinguished spaces C_p([1 , α]) , where α is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping T: C_p([1 , α]) → C_p(Y) is given. We also observe that for every countable ordinal α all closed linear subspaces of C_p([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: C_p(X) → C_k(X) _w, where C_k(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).