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Hill (Duke Math. J. vol. 9 (1942) p. 379), Agnew (Bull. Amer. Math. Soc. vol. 52 (1946) p. 128), and others have been concerned with finding conditions that summability methods should sum the convergent sequences and some others, or no others. We restrict ourselves to reversible matrix methods, that is, those in which the matrix A = (ank) transforms the sequence # = { # n } into the sequence Ax = {An(x)} = { X X o UnJeXic} and if y is any given convergent sequence, the equation y~Ax has a unique solution x. Our results are particularly simple in the special case of normal matrices, that is, those with dnk = 0 for k>n, ann5é0. We deal with conservative matrices, that is, those which transform every convergent sequence into a convergent sequence. The regular matrices form a subclass. Consider, for the moment, normal matrices. What is known, and indeed trivial, is the following: a normal conservative matrix A sums only convergent sequences if and only if A~ is conservative. Now, setting | |^4| |=supn y^Ln |#nfc|, A is conservative if and only if im i < oo and in addition the sequences formed from its columns, and the sequence { X X o #nfc}, are all convergent (Silverman-ToeplitzKojima-Carmichael et al.). Thus the necessary and sufficient conditions are transparently displayed. What we propose to do is reduce these conditions by two, keeping the one which the author has found easiest to check in certain special cases. According to Banach, Theorie des operations linéaires, Warsaw, 1932, p. SO, if A is reversible there is a bounded sequence \cu\ and a matrix B = (bnk) such that if y is convergent then y—Ax where # = { # n } , xn — cn limr yr+ X X o bnkyk, the series being always convergent. Clearly if A is normal, cn = 0 and B —A~ . As one of several possibilities let us define m _ 1 | | as supn {| cn\ + X X o | bnk\ }. This is consistent with the case when A is normal.

A necessary and sufficient condition that a summability method be stronger than convergence