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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the multiplicative behavior of regular matrices


Author: Robert E. Atalla
Journal: Proc. Amer. Math. Soc. 26 (1970), 437-446
MSC: Primary 47.25; Secondary 40.00
DOI: https://doi.org/10.1090/S0002-9939-1970-0271752-X
Addendum: Proc. Amer. Math. Soc. 48 (1975), 268.
MathSciNet review: 0271752
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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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DOI: https://doi.org/10.1090/S0002-9939-1970-0271752-X
Keywords: Regular matrix, $ C(X)$, multiplicative matrix method, invariant mean, $ \beta N$
Article copyright: © Copyright 1970 American Mathematical Society