An application of Banach linear functionals to summability

Abstract

1. A summability matrix is called conservative if it attaches a limit to, that is, sums, every convergent sequence. If moreover this limit is a fixed multiple m of the ordinary limit of the sequence the matrix is called multiplicative m. If a matrix A is multiplicative m then the matrix kA, where k $0 is a number, is multiplicative km and sums exactly the same sequences as A. Thus it is immaterial to specify m except to say whether or not it is zero. The dichotomy of matrices into those for which m =0, m S 0 is well known to be significant. A single example is the theorem of Steinhaus [1 ] (1) that if m #0 a multiplicative m matrix cannot sum all bounded sequences, the result being false if m = 0. The principal object of this paper is to extend this classification into the whole set of conservative matrices. This will be separated into the subclasses of co-regular and co-null matrices; the division of multiplicative matrices induced will be that mentioned above. It will then be shown that a class of theorems which have been proved about multiplicative matrices can be so generalized as to apply to conservative matrices in general. The value of the classification will appear in that certain results in which the condition m 0 plays an essential role will hold for co-regular but not for co-null matrices. Some results are new even for multiplicative matrices, for example, the specialization of Theorem 2.0.3. 1.1. Preliminary. Let A = (ac,k) be a matrix of complex numbers and x = {xn4 a complex sequence, then y =Ax is called the transformed sequence where in the multiplication x and y are treated as column vectors, thus Y= {yn} where yn= Z_o ankXk=An(x). Then, if it exists, lim yn=lim An(x) is called A(x). Finally the domain of the functional A(x), that is, the set of sequences x such that Ax is convergent, is called the summability field of A and written (A). We shall denote the identity matrix by I so that (I) is the set of convergent sequences. Setting, after Brudno (1), jAj =supn Ek= ank and denoting by F the set of sequences i= {I, 1, 1, . . . }, Io= {1, 0, 0, . . . }, 8'= {0, 1, 0, . . *} . . , Sk, * * * , we have the classical result: the matrix A is conservative if and only if (i) |jA|| < Xo and (ii) FC(A). Denoting A(Sk) = limn,O0 ank by ak we can easily show that