Banach space-valued random variables and tensor products of Banach spaces

Abstract

Since the publication in 1953 of Mourier’s paper [9] on random variables with values in a Banach space, a large number of papers concerned with the development of probability theory in Banach spaces have appeared. These papers have, in the main, been devoted to (i) measure-theoretic problems, (ii) limit theorems, (iii) martingale theory, and (iv) the theory of random equations. We refer to Bharucha-Reid [l], Driml and HanH [4], and Grenander [6] for references. Many studies in abstract probability theory and its applications lead to the spaces L,(Q, @, p, X) = L,(Q, X) o random variables defined on a probaf bility space (Q, ed, p) with values in a Bancah spaceX In this paper we utilize tensor product methods, as applied to the L,(Q, X) spaces, to consider a number of problems. In Section 2 we discuss (i) strong random variables, (ii) the spaces L&2, X) as tensor product Banach spaces L,(Q) B X, and (iii) the connections between the norms on the spaces L&2, X) and L&Q, X) and the y and h norms in the sense of tensor product crossnorms (cf. Schatten [14]). In Section 3 we utilize tensor product methods to prove the existence of the conditional expectation for strong random variables. Sections 4 and 5 are devoted to tensor product Hilbert spaces, and in Section 6 we consider a problem concerning the eigenvalues of a random Hermitian matrix. In Section 4 we show that every pair of elements of a tensor product Hilbert space H @ 8, where H and sj are Hilbert spaces,