Generalized bi-circular projections |
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Authors: | Pei-Kee Lin |
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Institution: | Department of Mathematics, University of Memphis, Memphis, TN 38152, USA |
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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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Keywords: | Isometry Generalized bi-circular projection |
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