G-invariant norms and bicircular projections

It is shown that for many finite dimensional normed vector spaces V over C, a linear projection P : V → V will have nice structure if P + λ(I − P) is an isometry for some complex unit not equal to one. From these results, one can readily determine the structure of bicircular projections, i.e., those linear projections P such that P + μ(I − P) is an isometry for every complex unit μ. The key ingredient in the proofs is the knowledge of the isometry group of the given norm. The proof techniques also apply to real vector spaces. In such cases, characterizations are given to linear projections P such that P − (I − P) = 2P − I is an isometry.