ABSTRACT
After considering isometries on vector-valued continuous function spaces, it is perhaps natural to turn to an investigation of isometries on vector-valued Lp spaces. Given a measure space (Ω,Σ, µ) and a Banach space X, the space Lp(µ, X) or just Lp(X) will mean the Banach space of (equivalence classes of) Bochner measurable functions F from Ω to X for which the norm
‖F‖p = {∫
‖F (t)‖pdµ }1/p
is finite. The norm ‖ · ‖ denotes the norm on the Banach space X. To say that F is Bochner measurable means that there exists a sequence {Fn} of simple functions such that limn ‖Fn(t) − F (t)‖ = 0 µ-almost everywhere. (A function F is sometimes called weakly µ-measurable if the scalar function x∗ ◦ F is measurable in the usual sense for each x∗ ∈ X∗.) By a simple function is meant a function of the form F =
∑n j=1 χAj xj , where A1, . . . , An
are measurable sets, χA is the characteristic function of A, and xj ∈ X for each j. Of course, in the case p = ∞ we have, as usual,
‖F‖∞ = ess sup‖F (t)‖. In this chapter we will assume that q is the extended real number conju-
gate to p, that is,
1 p
+ 1 q
= 1.
