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, A. Eysen, A. Leiderman and V. Valov, Linear and uniformly continuous surjections between $C_p$-spaces over metrizable spaces, Math. Slovaca, vol. 75 (2025), pp. 669–678, we obtained the following result:

Theorem. If $T: D_{p}(X) \to D_{p}(Y)$ 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 $\mathcal{P}$ from $X$.

A linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$ 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. In our talk, we establish sufficient conditions under which the above statement remains valid for a densely defined linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$. In particular, $\mathcal{P}$ can be zero-dimensionality, strong countable-dimensionality, or $\sigma$-compactness.

Additionally, for arbitrary Tychonoff spaces $X$ and $Y$, assuming only that $T: E_p(X)\to E_p(Y)$ is a densely defined linear continuous operator, we show that $X\in\mathcal P$ implies $Y\in\mathcal P$ where $\mathcal P$ is the property $(\kappa)$, the strong $\sigma$-scatteredness, or the property of being a $\Delta_1$-space.