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In operator theory, Atkinson's theorem (named for Frederick Valentine Atkinson) gives a characterization of Fredholm operators. == The theorem == Let ''H'' be a Hilbert space and ''L''(''H'') the set of bounded operators on ''H''. The following is the classical definition of a Fredholm operator: an operator ''T'' ∈ ''L''(''H'') is said to be a Fredholm operator if the kernel Ker(''T'') is finite-dimensional, Ker(''T *'') is finite-dimensional (where ''T *'' denotes the adjoint of ''T''), and the range Ran(''T'') is closed. Atkinson's theorem states: :A ''T'' ∈ ''L''(''H'') is a Fredholm operator if and only if ''T'' is invertible modulo compact perturbation, i.e. ''TS'' = ''I'' + ''C''1 and ''ST'' = ''I'' + ''C''2 for some bounded operator ''S'' and compact operators ''C''1 and ''C''2. In other words, an operator ''T'' ∈ ''L''(''H'') is Fredholm, in the classical sense, if and only if its projection in the Calkin algebra is invertible. 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「Atkinson's theorem」の詳細全文を読む スポンサード リンク
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