\(G\)-invariant norms and bicircular projections (Q861026)

It is shown that, for many finite dimensional normed linear spaces \(V\) over \({\mathbb C}\), a linear projection \(P:V\to V\) has a nice structure if \(P+\lambda (I-P)\) is an isometry for some complex unit not equal to one. The results are obtained for spaces equipped with the following norms: symmetric, unitarily invariant, unitary congruence invariant, and \(G\)-invariant norms, where \(G\) is a group of linear operators on~\(V\) of the form \(A\mapsto U^*AU\) for some \(U\in U({\mathbb C}^n)\). From these results, one can readily determine the structure of bicircular projections, i.e., those linear projections \(P\) such that \(P+\mu(I-P)\) is an isometry for every complex unit~\(\mu\). The key ingredient in the proof is the knowledge of the isometry group of the given norm. The proof technique is also appicable to spaces over~\({\mathbb R}\). In such cases, characterizations are given to linear projections \(P\) such that \(P-(I-P)=2P-I\) is an isometry.