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

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