Abstract
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.
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