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\(G\)-invariant norms and bicircular projections. (English) Zbl 1110.15025

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.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
46B04 Isometric theory of Banach spaces
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