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In operator theory, a discipline within mathematics, the Wold decomposition, named after Herman Wold, or Wold–von Neumann decomposition, after Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator. In time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. == Details == Let ''H'' be a Hilbert space, ''L''(''H'') be the bounded operators on ''H'', and ''V'' ∈ ''L''(''H'') be an isometry. The Wold decomposition states that every isometry ''V'' takes the form : for some index set ''A'', where ''S'' in the unilateral shift on a Hilbert space ''Hα'', and ''U'' is a unitary operator (possible vacuous). The family consists of isomorphic Hilbert spaces. A proof can be sketched as follows. Successive applications of ''V'' give a descending sequences of copies of ''H'' isomorphically embedded in itself: : where ''V''(''H'') denotes the range of ''V''. The above defined . If one defines : then : It is clear that ''K''1 and ''K''2 are invariant subspaces of ''V''. So ''V''(''K''2) = ''K''2. In other words, ''V'' restricted to ''K''2 is a surjective isometry, i.e., a unitary operator ''U''. Furthermore, each ''Mi'' is isomorphic to another, with ''V'' being an isomorphism between ''Mi'' and ''M''''i''+1: ''V'' "shifts" ''Mi'' to ''M''''i''+1. Suppose the dimension of each ''Mi'' is some cardinal number ''α''. We see that ''K''1 can be written as a direct sum Hilbert spaces : where each ''Hα'' is an invariant subspaces of ''V'' and ''V'' restricted to each ''Hα'' is the unilateral shift ''S''. Therefore : which is a Wold decomposition of ''V''. 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「Wold's decomposition」の詳細全文を読む スポンサード リンク
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