Bilinear transformations in Hilbert space

Abstract

Introduction. A function of two variables h = F(f, g), where h, f, and g are all elements of Hilbert Space may be termed a bilinear transformation if it is linear in f and linear in g. A more formal definition is given in ?1. While a complete treatment of bilinear transformations would obviously require a very lengthy discussion, we wish to point out in this paper that many of the methods used in the study of linear transformations are applicable to them, with, of course, certain modifications. Many elementary notions can be extended and corresponding results obtained. For certain classes of bilinear transformations, there is even a canonical resolution (cf. ?5, Theorem 7). Bilinear transformations have appeared in the work of Kerner. t While the first Frechet differential is a linear transformation, the second is bilinear, and it is this connection which was studied by Kerner. We shall show the relationship between bilinear transformations and rings of operators.: Mazur and Orlicz have pointed out the relationship between bilinear (and multilinear) transformations and polynomial transformations (cf. [5 1, p. 59). Polynomial transformations have also been studied by Banach (cf. [2]). We shall have occasion to use some of their results. There is a very simple relationship between bilinear transformations and trilinear forms. For instance, if F(f, g) is a bilinear transformation, then