Commutants for some classes of Hausdorff matrices

Abstract:

Let $\Gamma$ denote the algebra of all bounded infinite matrices on c, the space of convergent sequences, $\Delta$ the subalgebra of $\Gamma$ consisting of lower triangular matrices. It is well known that, if H is any Hausdorff matrix with distinct diagonal entries, then the commutant of H in $\Delta$ contains only Hausdorff matrices. In previous work the author has shown that a necessary condition for the commutant of a Hausdorff matrix H to be the same in $\Gamma$ and $\Delta$ is that H have distinct diagonal entries, but that the condition is not sufficient. In this paper it is shown that certain Hausdorff matrices, with distinct diagonal entries, have the same commutants in $\Gamma$ and $\Delta$.