A Banach Algebra Criterion for Tauberian Theorems

We acknowledge valuable discussions with Dr. J. Lindenstrauss of Yale University. An infinite matrix A is called conservative if for each convergent sequence x the transformed sequence Ax is convergent. It is a classical theorem that a necessary and sufficient condition that a matrix A be conservative is that: (1) \\A\\ =supi Ey" i \aa\ < °°> (2) lim,-,«, EyTM iaa exists, (3) lim,-,M an exists for each i. It is well known that the set of conservative matrices forms a Banach algebra; we will call this Banach algebra T. Throughout this paper "matrix" will mean "conservative matrix" unless otherwise stated. Many criteria for a matrix to sum no bounded divergent sequence are known; cf. Copping [l], Wilansky and Zeller [2], [4]. The purpose of this paper is to show that if A is not a left divisor of zero (abbreviated l.z.d.) in V then a necessary and sufficient condition that A sum some bounded divergent sequence is that A be a left topological divisor of zero (abbreviated l.t.z.d.) inT (i.e. for e>0 there exists BET such that ||y3|| = 1 and ||^4-B|| <«). Equivalently, a necessary and sufficient condition that A, A not a l.z.d., sum a bounded divergent sequence is the following: For e>0 there exists a convergent sequence, x, such that ||x|| = sup„ |x(«)| =1 and \\Ax\\ n) conservative matrix not a l.z.d. sums a bounded divergent sequence if it is on the boundary of the maximal group. We shall show that this theorem is true even if "triangular" is omitted. As Copping [l] shows, the converse is false. Hence there are l.t.z.d.'s in the Banach algebra, A, of conservative triangular matrices not on the boundary of the maximal group. However the structure of the set of l.t.z.d. 's in both A and T still eludes us.