A Tauberian Theorem for Stretchings

R. C. Buck fl] has shown that if a regular matrix sums every subsequence of a sequence x, then x is convergent. I. J. Maddox [4] improved Buck's theorem by showing that if a non-Schur matrix sums every subsequence of a sequence x, then x is convergent. Actually Maddox proved a stronger result: If x is divergent and A sums every subsequence of x, then A is a Schur matrix, i.e., It should be remarked that a matrix is a Schur matrix if and only if it sums every bounded sequence. In [3] we worked with stretchings of sequences, and obtained a result similar to Buck's theorem. The purpose of the present paper is to give a result on stretchings which is analogous to Maddox's theorem. Thus we have the THEOREM. / / x is a divergent complex sequence and A is a complex matrix which sums every stretching of x, then there exists N such that (i) a pq-> c q as p-» oo, q > N, 00 (ii) £ c q converges, q=N+l oo (iii) £ (a pq-c q)-> 0 as p-» oo. If D denotes the set of all matrices A such that for some N(A), (i), (ii), and (iii) hold for N = N(A), then our theorem can be formulated as a Tauberian theorem: If a non-D matrix sums every stretching of a sequence x, then x is convergent. We begin with a lemma which may be of independent interest. LEMMA. / / x is a sequence and A is a matrix with each Z a pq q = l convergent, then there exist a stretching x* of x and a row finite matrix A* such that if y is any stretching of x*, then Ay exists and Ay and A*y are either both convergent or