Generalizations of Banach-hausdorff Limits

In a recent paper [l],1 W. F. Eberlein introduced the notion of Banach-Hausdorff limits. We employ throughout m to denote the space of all real bounded sequences [2, pp. 11 and 34]. The BanachHausdorff limits are real-valued functionals L(x), defined over m, which are Banach limits [2, p. 34], i.e., which satisfy the four conditions (i) L(ax+by)=aL(x)+bL(y) (a, b real), (ii)L(l) = l, (iii) L(x)^0 if x^O (i.e., if x„^0 for all n), (iv) L(Sx) =L{x), where 5 is the "translation" matrix s„,B+i=l (« = 1,2, • • • ), s„,jfc = 0 (k^n + 1), so that S(xn)=xn+i, and which, in addition to being Banach limits, satisfy the condition (v) L{Hx)=L(x),HEK+, where 3C+ is the semi-group of non-negative regular Hausdorff matrices,2 i.e., the set of T-matrices [3, pp. 64-65] given by