Linear continuous operators with bounded supports

Abstract

For any Tychonoff space $X$ let $D(X)$ be either the set $C(X)$ of all continuous functions on $X$ or the set $C^{\ast }(X)$ of all bounded continuous functions on $X$. When $D(X)$ is endowed with the point convergence topology, we write $D_p(X)$. Zakrzewski \cite[Theorem 3.12]{kz} proved that if $X$ and $Y$ are $\ddot{I}ƒ$-compact spaces and there is a continuous linear map $T:C_p(X)\to C_p(Y)$ such that $T(C_p(X))$ is dense in $C_p(Y)$ and $|\supp(y)|\leq m$ for every $y\in Y$, then $\dim Y\le m\cdot \dim X+m+m!-1$. Here, $\supp(y)$ denotes the support of the linear continuous map $l_y:C_p(X)\to \mathbb{R}$, defined by $l_y(f)=T(f)(y)$. In the present paper we improve the last inequality by showing that $\dim Y\le m\cdot \dim X$ provided $X,Y$ are Tychonoff spaces and there is a continuous linear surjection $T:D_p(X)\to D_p(Y)$ with $|\supp(y)|\leq m$ for every $y\in Y$. This implies the following generalization of \cite[Theorem 1.4]{ev}: If $T:D_p(X)\to D_p(Y)$ is a continuous linear surjection with $X,Y$ Tychonoff spaces and $\dim X=0$, then $\dim Y=0$. Our proofs are obtained by refining the techniques developed in \cite{ev}.

18 pages