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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク normal operator ： ウィキペディア英語版 normal operator In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N *'', that is: ''NN *'' = ''N *N''. Normal operators are important because the spectral theorem holds for them. The class of normal operators is well-understood. Examples of normal operators are * unitary operators: ''N *'' = ''N−1 * Hermitian operators (i.e., self-adjoint operators): ''N *'' = ''N'' * Skew-Hermitian operators: ''N *'' = −''N'' * positive operators: ''N'' = ''MM *'' for some ''M'' (so ''N'' is self-adjoint). A normal matrix is the matrix expression of a normal operator on the Hilbert space C''n''. == Properties == Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable. Let ''T'' be a bounded operator. The following are equivalent. *''T'' is normal. *''T *'' is normal. *||''Tx''|| = ||''T *x''|| for all ''x'' (use $\backslash |Tx\backslash |^2\; =\; \backslash langle\; T^$ *Tx, x \rangle = \langle TT^ *x, x \rangle = \|T^ *x\|^2). *The selfadjoint and anti-selfadjoint parts of ''T'' commute. That is, if we write $T\backslash equiv\; T\_1+i\; T\_2$ with $T\_1:=\backslash frac$ and $i\backslash ,T\_2:=\backslash frac,$, then $T\_1T\_2=T\_2T\_1$.〔In contrast, for the important class of Creation and annihilation operators of, e.g., quantum field theory, they don't commute〕 If ''N'' is a normal operator, then ''N'' and ''N *'' have the same kernel and range. Consequently, the range of ''N'' is dense if and only if ''N'' is injective. Put in another way, the kernel of a normal operator is the orthogonal complement of its range. It follows that the kernel of the operator ''Nk'' coincides with that of ''N'' for any ''k''. Every generalized eigenvalue of a normal operator is thus genuine. λ is an eigenvalue of a normal operator ''N'' if and only if its complex conjugate $\backslash overline$ is an eigenvalue of ''N *''. Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and a normal operator stabilizes the orthogonal complement of each of its eigenspaces. This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator. There is also an infinite-dimensional version of the spectral theorem expressed in terms of projection-valued measures. The residual spectrum of a normal operator is empty.〔 The product of normal operators that commute is again normal; this is nontrivial, but follows directly from Fuglede's theorem, which states (in a form generalized by Putnam): :If $N\_1$ and $N\_2$ are normal operators and if ''A'' is a bounded linear operator such that $N\_1\; A\; =\; A\; N\_2$, then $N\_1^$ * A = A N_2^ *. The operator norm of a normal operator equals its numerical radius and spectral radius. A normal operator coincides with its Aluthge transform. 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「normal operator」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.