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Anticommuting selfadjoint operators. (English) Zbl 0704.47020

Author’s summary: Several equivalent definitions of anticommutativity for self-adjoint operators are presented. It is shown that the anticommutativity of two self-adjoint operators A and B, satisfying \(<Af| Bg>+<Bf| Ab>=0\) for all f and g in the intersection of the domains of A and B, is equivalent to the self-adjointness of their sum and difference. In studying the structure of anticommuting self-adjoint operators A and B, attention may be restricted to the case where both A and B are injective. In that case, A is unitarily equivalent to -A. Moreover, A and B are uniquely determined by a triplet (b,P,Q), where b is unitary, and P and Q are commuting, injective, positive, and self- adjoint operators. This is a natural correspondence.
Reviewer: B.P.Duggal

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47B47 Commutators, derivations, elementary operators, etc.
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References:

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