Groups of Isometries on Banach Spaces

In this section, we describe the general framework for the spectral analysis of groups of isometrics on Banach spaces.

Let be a Banach space and a closed linear subspace. Besides the norm topologies, we shall also consider the weak topologies on and on. Consider the following conditions on the pair:

Lemma 1. Let be a pair-satisfying conditions and a bounded regular Borel measure on a separable locally compact Hausdorff space S. For every w-continuous norm-bounded function x there exists a unique element such that

Proof. The equation

defines a linear form on. To prove the lemma it is sufficient to show that is - continuous or, equivalently, that f is continuous with respect to the Mackey topology. Thus, we have to show that there exist an absolutely convex w-compact set and such that

We first assume that C = supp is compact. Then we have

Since C is compact and is w-continuous, the set is w-compact. Then the set is w-compact, and hence is absolutely convex and w-compact by condition (2). Since, inequality (2) follows from (3).

In the general case, there exists an increasing sequence of compact subsets of S such that. By the first part or the proof, there exist such that

Then, for every we have

and using condition it follows that is a Cauchy sequence in. If is the limit of this sequence, it then follows from (4) that for all .

The uniqueness of the element satisfying (1) follows obviously using (1x).

The unique element satisfying (1) will be denoted by

Consider now two pairs and satisfying conditions and. Let be the Banach space or all bounded linear operators and the linear space or all w-continuous linear operators. Using the Banach–Steinhauss theorem, it is easy to check that is a norm-closed linear subspace. In particular, is a Banach space.