Topological Divisors of Zero and Tauberian Theorems
Albert Wilansky · Transactions of the American Mathematical Society · 1964
1. Introduction.The connection between the two topics mentioned in the title has been shown by I. D. Berg [2].(See Theorem 6, below.)The author thanks Dr. Berg for assistance with the material of this article.The topological divisors of 0 in the algebra £[X] of all bounded endomorphisms of a Banach space X are fairly well understood [15; 22].In this article we shall extend this knowledge to a subalgebra of B[X], and deduce consequences in the form of Tauberian theorems.Algebraists will find no very deep algebraic results in the extension, but they may be interested to see how algebraic properties correspond to analytic concepts.The very oldest such correspondence is undoubtedly the remark that if a transformation is invertible in the algebraic sense, it is trivial in the summability sense in that it carries no divergent sequences into convergent ones.2. Notation.All the notation given, except for \¡/ and %", is standard, and may be found, for example, in [14; 15; 18; 19].By c0, c, m are meant, respectively, the spaces of null, convergent and bounded sequences x = {x"}, n =1,2,--.For xec0, c, m, ¡x|| = sup|x"|.The space of sequences x with Z | x" | < co is written /.The constant sequence of ones is written 1, and ô", n = 1,2,---, is the sequence whose nth term is 1, all other terms 0.If A = ia"k), n, fc= 1,2,-", is a matrix, Ax is defined to be {(Ax)"} where (-4x)" = Z*"L i ankxk, and x is called summable by A ifx e cA where cA = {x:Axec}, and lim/lx is also written lim^x.If cA Z3C, A is called conservative.If lim^x = limx for all xec, A is called permanent.(We avoid the more usual term "regular" which has a quite different significance in algebra.For the same reason we deplore the use of the word "normal" to describe the shape of certain matrices.)If limAx = t limx for all xec, A is called multiplicative-t.B[X] stands for the Banach algebra of endomorphisms (always assumed bounded) of any Banach space X, with || T|| = sup{| Tx || : || x || ^ 1}, and X' is the space of continuous linear functionals on X.An endomorphism which is one-to-one and onto is called an automorphism.