Linear and uniformly continuous surjections between C_p-spaces over metrizable spaces

For any Tychonoff space X let D(X) be either the set C(X) of all continuous functions on X or the set C^∗(X) of all bounded continuous functions on X. When D(X) is endowed with the pointwise convergence topology, we write D_p(X). Let T: D_p(X) → D_p(Y) be a continuous linear surjection, where X is a metrizable space and Y is perfectly normal. We show that if X has some dimensional-like property, then so does Y. For example, could be one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. This result remains true if T is a uniformly continuous and inversely bounded surjection. Also, we consider other properties: of being a scattered space, or a strongly σ-scattered space, or a Δ1-space. Our results strengthen and extend several results from the various recently published papers.