Infinite matrices and invariant means

Abstract

Let a be a one-to-one mapping of the set of positive integers into itself such that av(n)On for all positive integers n and p, where aP(n) = a(a-1(n)), p = 1, 2, . A continuous linear functional p on the space of real bounded sequences is an invariant mean if p(x)_O when the sequence x={x,} has xn_O for all n, p({l, 1, 1, * })=+1, and T({x,(n)})=P(x) for all bounded sequences x. Let VG be the set of bounded sequences all of whose invariant means are equal. If A = (ank) is a real infinite matrix, then A is said to be (1) a-conservative if Ax={Y2k ankXk} C VI for all convergent sequences x, (2) a-regular if Ax E VG and p(Ax)=lim x for all convergent sequences x and all invariant means 9, and (3) a-coercive if Ax E VG for all bounded sequences x. Necessary and sufficient conditions are obtained to characterize these classes of matrices.