On densely defined linear continuous operators between function spaces
Abstract
For any Tychonoff space X, let D(X) denote either the space C(X) of all continuous real-valued functions on X or the space C*(X) of all bounded continuous real-valued functions on X. We write Dp(X) when D(X) is endowed with the topology of pointwise convergence. In our recently published paper [8, Theorem 1.6], we obtained the following result: if T: Dp(X) → Dp(Y) is a linear continuous surjection, where X is a metrizable space and Y is a perfectly normal space, then Y inherits a given topological property 𝒫 from X. A linear continuous surjection T: Ep(X)→ Ep(Y) is said to be densely defined if E(X) and E(Y) are dense linear subspaces of Dp(X) and Dp(Y), respectively, and E(X) is correct (see Definition 1.5(a)). In the present paper, we establish sufficient conditions under which the above statement remains valid for a densely defined linear continuous surjection T: Ep(X) → Ep(Y). In particular, 𝒫 can be zero-dimensionality, σ-compactness or strong countable-dimensionality. Additionally, for arbitrary Tychonoff spaces X and Y, assuming that T: Ep(X)→ Ep(Y) is a densely defined linear continuous operator, we show that X∈𝒫 implies Y∈𝒫, where 𝒫 is the property (κ), the strong σ-scatteredness, or the property of being a Δ1-space.
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BibTeX
@misc{leiderman2026densely,
title = {{On densely defined linear continuous operators between function spaces}},
author = {Arkady Leiderman and Vesko Valov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37271},
eprint = {2609.37271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37271},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On densely defined linear continuous operators between function spaces
AU - Arkady Leiderman
AU - Vesko Valov
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37271
UR - https://arxiv.org/abs/2609.37271
AB - For any Tychonoff space X, let D(X) denote either the space C(X) of all continuous real-valued functions on X or the space C^*(X) of all bounded continuous real-valued functions on X. We write D_p(X) when D(X) is endowed with the topology of pointwise convergence. In our recently published paper [8, Theorem 1.6], we obtained the following result: if T: D_p(X) → D_p(Y) is a linear continuous surjection, where X is a metrizable space and Y is a perfectly normal space, then Y inherits a given topological property 𝒫 from X. A linear continuous surjection T: E_p(X)→ E_p(Y) is said to be densely defined if E(X) and E(Y) are dense linear subspaces of D_p(X) and D_p(Y), respectively, and E(X) is correct (see Definition 1.5(a)). In the present paper, we establish sufficient conditions under which the above statement remains valid for a densely defined linear continuous surjection T: E_p(X) → E_p(Y). In particular, 𝒫 can be zero-dimensionality, σ-compactness or strong countable-dimensionality. Additionally, for arbitrary Tychonoff spaces X and Y, assuming that T: E_p(X)→ E_p(Y) is a densely defined linear continuous operator, we show that X∈𝒫 implies Y∈𝒫, where 𝒫 is the property (κ), the strong σ-scatteredness, or the property of being a Δ_1-space.
ER -