Commutants for some classes of Hausdorff matrices

Abstract

Let Γ \Gamma denote the algebra of all bounded infinite matrices on c, the space of convergent sequences, Δ \Delta the subalgebra of Γ \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 Δ \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 Γ \Gamma and Δ \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 Γ \Gamma and Δ \Delta .