A concrete realization of the dual space of L¹-spaces of certain vector and operator-valued measures

Abstract

For a given vector measure n, an important problem, but in practice a difficult one, is to give a concrete description of the dual space of L l (n). In this note such a description is presented for an important class of measures n, namely the spectral measures (in the sense of N. Dunford) and certain other vector and operator-valued measures that they naturally induce. The basic idea is to represent the 1} -spaces of such measures as a more familiar space whose dual space is known.

Non-weak compactness of the integration map for vector measures

“…The identity (7), which actually holds for all E e X and not just Q, follows from Lemma 2.3 and the identities (8) […”

“…To establish this claim we need the following result; the notation is as in Example 2. [ 8 ] . REMARK 3 (b).…”

“…For measures m with bounded variation the space L (m)' was considered in [4] where, unfortunately, the main result is false. For the case of vector measures induced by spectral measures via evaluation, a concrete description of L (m)' is given in [8]. One description of the individual elements of L l (m)' for an arbitrary vector measure m is presented in [6].…”

“…The identity (7), which actually holds for all E e X and not just Q, follows from Lemma 2.3 and the identities (8) [ PROOF. We note that h e L\X) is non-negative if, and only if, (h, £) = J a £hdX > 0, for all £ e L°°(X) such that £ > 0.…”

“…That m is not an evaluation of some spectral measure at a point of X now follows from Lemma 3.3 and [6; Theorem 10], which states that the integration map /^Z -t Z (of a Banach space valued vector measure v: Z -• Z) is an isomorphism onto a closed subspace of Z if and only if there exists a closed subspace Y of Z , a closed spectral measure P: 2 -» L S (Y) and a vector y eY such that v = J o Py (where J is the natural inclusion of Y into Z ) . Of course, one also needs to use the fact that the integration map I p : L x {Py) -* Y is an isomorphism of L l (Py) onto the closed linear span of {P(E)y; £ e l } , [ 8 ] .…”