Abstract
Given two orthogonal projections P and Q, we are interested in all unitary operators U such that \(UP=QU\) and \(UQ=PU\). Such unitaries U have previously been constructed by Wang, Du, and Dou and also by one of the authors. One purpose of this note is to compare these constructions. Very recently, Dou, Shi, Cui, and Du described all unitaries U with the required property. Their proof is via the two projections theorem by Halmos. We here give a proof based on the supersymmetric approach by Avron, Seiler, and one of the authors.
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Research of Barry Simon supported in part by NSF Grants DMS-1265592 and DMS-1665526 and in part by Israeli BSF Grant No. 2014337. Ilya Spitkovsky was supported in part by Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi.
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Böttcher, A., Simon, B. & Spitkovsky, I. Similarity Between Two Projections. Integr. Equ. Oper. Theory 89, 507–518 (2017). https://doi.org/10.1007/s00020-017-2414-6
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DOI: https://doi.org/10.1007/s00020-017-2414-6