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Commutation properties of the partial isometries associated with anticommuting self-adjoint operators

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Please use this identifier to cite or link to this item:http://doi.org/10.14943/83266

Title: Commutation properties of the partial isometries associated with anticommuting self-adjoint operators
Authors: Arai, Asao Browse this author
Issue Date: Aug-1991
Journal Title: Hokkaido University Preprint Series in Mathematics
Volume: 121
Start Page: 2
End Page: 25
Abstract: It is proven that, for every pair {A, B} of anticommuting self-adjoint operators, iAB is essntially self-adjoint on a suitable domain and its closure O(A, B) anticommutes with A and B. For every self-adjoint opearlor S, a partial isometry Us is defined by the polar decomposition S = Us lSI. Let Ps be the orthogonal projection onto (Ker S)l. . The commutation properties of' the operators UA, UB, Uc(A,B), PA , PB , and PAPB are investigated. These operators multiplied by some constants satisfy a set of' commutation, relations, which may be regarded as an extension of that satisfied by the standard basis of the Lie algebra .au(2, C) of' the special unitary group SU(2). It is shown that there exists a Lie algebra ro? associated with those operators and that, if' A and B are injective, then ro? gives a completely reducible representation of su(2, C) with the heighest weight of' each irreducible component being 1/2. Moreover, the "diagonalization" of' A+ B is given.
Type: bulletin (article)
URI: http://hdl.handle.net/2115/68868
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

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