Addenda to the Paper on Bocher's Theorem
Aurel Wintner · American Journal of Mathematics · 1956
If (aik) a ==a(t), where 0 ? t 1, then a C R is neither necessary nor sufficient for a C S (cf. the paper quoted in the title, which will be referred to as loc. cit.). On the other hand. if a C L means (as usual, with L =LI) that y e R holds for the norm, -y = I a J, of a also, then, according to a criterion which goes back to Bocher, a C L is sufficient for a C S (for references to the implication L > S, and for generalizations of L > S derived from L =>S itself, cf. loc. cit.). There is, however, something unsatisfactory in the standard criterion, L =>S. In fact, this criterion fails to contain the fact that, as readily seen from a quadrature, a C R is sufficient (and necessary) for a C S in the scalar case, n1 1. It is therefore of initerest that L > S can be improved to the following criterion: If ao (t) denotes the diagonal matrix the diagonal elements of which are those of a(t), then aO C R and a -aO L together are sufficient for a C S. It will be clear from the following proof that (and in which maIiner) this extended criterion can be combined with the criteria giveni loc. cit. First, it follows from L ) S by a known application of the method of the variation of constants (cf. vol. 68 (1946), p. 200, of this Journal), that if 3 =, /B(t) is any continuous matrix the real part of the trace of which has an indefinite integral posessing a lower bound (> -oo), then /3 C S and a -,B C L suffice for a C S. Since the trace condition is certainly satisfied if ,/ C R, it follows, by choosing / =ao, that the italicized assertion is proved