On the Multiplicative Behavior of Regular Matrices
Robert E. Atalla · Proceedings of the American Mathematical Society · 1970
Let $T$ be a bounded linear operator on $C(X),X$ compact ${T_2}$, with $T1 = 1$. We define ${M_T}$ to be the subalgebra of $C(X)$ consisting of $g$ such that $T(fg) = TfTg$ for all $f$, and give a characterization of ${M_T}$. We apply the characterization to the multiplicative behavior of regular matrices, considering these as linear operators on $C(\beta N\backslash N)$. We also relate invariance properties of a matrix under suitable mappings of the integers to topological properties of its support set in $\beta N\backslash N$, and give an example of a nonnegative multiplicative matrix whose support set is nowhere dense in $\beta N\backslash N$.