Regular Methods of Summability and the Banach-Saks Property for Double Sequences

A Banach space B is said to satisfy the Banach-Saks property with respect to a regular summability method if every bounded subsequence has a summable subsequence. We show that if a Banach space satisfies the Banach-Saks property with respect to a Robison-Hamilton regular summability method, for every bounded double sequence there exists a beta-subsequence whose subsequences are all summable to the same limit.