On the multiplicative behavior of regular matrices

Atalla, Robert E.

doi:10.1090/S0002-9939-1970-0271752-X

On the multiplicative behavior of regular matrices
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Proc. Amer. Math. Soc. 26 (1970), 437-446 Request permission

Addendum: Proc. Amer. Math. Soc. 48 (1975), 268.

Abstract:

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 $\beta N\backslash N$.

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Multiplicative summability methods and the Stone-Čech compactification

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