Generalized bi-circular projections

Recall that a projection P on a complex Banach space X is a generalized bi-circular projection if P +λ(I −P) is a (surjective) isometry for some λ such that |λ| = 1 and λ = 1. It is easy to see that every hermitian projection is generalized bi-circular. A generalized bi-circular projection is said to be nontrivial if it is not hermitian. Botelho and Jamison showed that a projection P on C([0, 1]) is a nontrivial generalized bi-circular projection if and only if P − (I − P) is a surjective isometry. In this article, we prove that if P is a projection such that P + λ(I − P) is a (surjective) isometry for some λ, then either P is hermitian or λ is an nth unit root of unity. We also show that for any nth unit root λ of unity, there are a complex Banach space X and a nontrivial generalized bi-circular projection P on X such that P +λ(I −P) is an isometry. © 2007 Elsevier Inc. All rights reserved