On the Spectra of Group Commutators
نویسندگان
چکیده
1. All operators will be supposed bounded, linear (everywhere defined) over a Hilbert space. By the commutator, C, of two operators A and B will be meant the operator (1) C = AB BA. A conjecture of Kaplansky, cited in [4], that if A commutes with C, so that (2) AC = CA, then (3) sp(C) = 0 only (sp = spectrum), was recently proved by Kleinecke [6] and Shirokov [9]. Earlier proofs of (3) under the additional restriction that B also commutes with C were given in [8] and [lO]. In case A and B are nonsingular, so that 0 does not belong to the spectrum of either A or B, one can consider not only the "ring" commutator (1) but also the "group" commutator D defined by (4) D = ABA~lB-\
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