Extremal invariant subspaces (Q1061977)

Let \(G\) be a locally compact group, \(H\) a topological vector space and \(T\) a representation of \(G\) in \(H\). Coarsening the situation a little, one may say that the paper deals with the problem of existence of the weakest and the strongest norm among all \(T\)-invariant norms defined on various \(T\)-invariant subspaces of \(H\). It is proved that under some conditions the extremal norms do exist. Examples are given in which they are calculated explicitely, the principal one being the natural action of the group of analytic automorphisms of the unit ball of \({\mathbb{C}}^ n\) in the space of analytic functions on this ball. For \(n=1\) this contains the main result of the paper [\textit{L. A. Rubel}, \textit{R. M. Timoney}, Proc. Am. Math. Soc. 75, 45-49 (1979; Zbl 0405.46020)].