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In mathematics, the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of operators in the ideal with bounded operators. Modern characterisation of the commutator subspace is through the Calkin correspondence and it involves the invariance of the Calkin sequence space of an operator ideal to taking Cesàro means. This explicit spectral characterisation reduces problems and questions about commutators and traces on two-sided ideals to (more resolvable) problems and conditions on sequence spaces. == History == Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured in the matrix mechanics, or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician Paul Halmos in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators.〔 〕 In 1971 Carl Pearcy and David Topping revisited the topic and studied commutator subspaces for Schatten ideals.〔 〕 As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of Hilbert–Schmidt operators.〔 〕〔 〕 British mathematician Nigel Kalton, noticing the spectral condition of Weiss, characterised all trace class commutators.〔 〕 Kalton's result forms the basis for the modern characterisation of the commutator subspace. In 2004 Ken Dykema, Tadeusz Figiel, Gary Weiss and Mariusz Wodzicki published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.〔 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「commutator subspace」の詳細全文を読む スポンサード リンク
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