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In operator theory, a discipline within mathematics, a bounded operator ''T'': ''X'' → ''Y'' between normed vector spaces ''X'' and ''Y'' is said to be a contraction if its operator norm ||''T''|| ≤ 1. Every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias. == Contractions on a Hilbert space == If ''T'' is a contraction acting on a Hilbert space , the following basic objects associated with ''T'' can be defined. The defect operators of ''T'' are the operators ''DT'' = (1 − ''T *T'')½ and ''DT *'' = (1 − ''TT *'')½. The square root is the positive semidefinite one given by the spectral theorem. The defect spaces and are the ranges Ran(''DT'') and Ran(''DT *'') respectively. The positive operator ''DT'' induces an inner product on . The inner product space can be identified naturally with Ran(''D''''T''). A similar statement holds for . The defect indices of ''T'' are the pair : The defect operators and the defect indices are a measure of the non-unitarity of ''T''. A contraction ''T'' on a Hilbert space can be canonically decomposed into an orthogonal direct sum : where ''U'' is a unitary operator and Γ is ''completely non-unitary'' in the sense that it has no reducing subspaces on which its restriction is unitary. If ''U'' = 0, ''T'' is said to be a completely non-unitary contraction. A special case of this decomposition is the Wold decomposition for an isometry, where Γ is a proper isometry. Contractions on Hilbert spaces can be viewed as the operator analogs of cos θ and are called operator angles in some contexts. The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices. 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「Contraction (operator theory)」の詳細全文を読む スポンサード リンク
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