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| {{otheruses4|the general mathematical result|the application to time series analysis|Wold's theorem}}
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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 [[isometry|isometric linear operator]]s on a given [[Hilbert space]]. It states that every isometry is a direct sums of copies of the [[unilateral shift]] and a [[unitary operator]].
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| In [[time series analysis]], the theorem implies that any [[Stationary process|stationary]] discrete-time [[stochastic process]] can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a [[moving average process]].
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| == Details ==
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| 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
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| :<math>V = (\oplus_{\alpha \in A} S) \oplus U</math>
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| for some index set ''A'', where ''S'' in the [[unilateral shift]] on a Hilbert space ''H<sub>α</sub>'', and ''U'' is an unitary operator (possible vacuous). The family {''H<sub>α</sub>''} consists of isomorphic Hilbert spaces.
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| A proof can be sketched as follows. Successive applications of ''V'' give a descending sequences of copies of ''H'' isomorphically embedded in itself:
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| :<math>H = H \supset V(H) \supset V^2 (H) \supset \cdots = H_0 \supset H_1 \supset H_2 \supset \cdots, </math>
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| where ''V''(''H'') denotes the range of ''V''. The above defined <math>H_i = V^i(H)</math>. If one defines | |
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| :<math>M_i = H_i \ominus H_{i+1} = V^i (H \ominus V(H)) \quad \text{for} \quad i \geq 0 \;,</math>
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| then
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| :<math>H = (\oplus_{i \geq 0} M_i) \oplus (\cap_{i \geq 0} H_i) = K_1 \oplus K_2.</math>
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| It is clear that ''K''<sub>1</sub> and ''K''<sub>2</sub> are invariant subspaces of ''V''.
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| So ''V''(''K''<sub>2</sub>) = ''K''<sub>2</sub>. In other words, ''V'' restricted to ''K''<sub>2</sub> is a surjective isometry, i.e. an unitary operator ''U''.
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| Furthermore, each ''M<sub>i</sub>'' is isomorphic to another, with ''V'' being an isomorphism between ''M<sub>i</sub>'' and ''M''<sub>''i''+1</sub>: ''V'' "shifts" ''M<sub>i</sub>'' to ''M''<sub>''i''+1</sub>. Suppose the dimension of each ''M<sub>i</sub>'' is some cardinal number ''α''. We see that ''K''<sub>1</sub> can be written as a direct sum Hilbert spaces
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| :<math>K_1 = \oplus H_{\alpha}</math>
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| where each ''H<sub>α</sub>'' is an invariant subspaces of ''V'' and ''V'' restricted to each ''H<sub>α</sub>'' is the unilateral shift ''S''. Therefore
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| :<math>V = V \vert_{K_1} \oplus V\vert_{K_2} = (\oplus_{\alpha \in A} S) \oplus U,</math>
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| which is a Wold decomposition of ''V''.
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| === Remarks ===
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| It is immediate from the Wold decomposition that the [[spectrum (functional analysis)|spectrum]] of any proper, i.e. non-unitary, isometry is the unit disk in the complex plane.
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| An isometry ''V'' is said to be '''pure''' if, in the notation of the above proof, ∩<sub>''i''≥0</sub> ''H''<sub>''i''</sub> = {0}. The '''multiplicity''' of a pure isometry ''V'' is the dimension of the kernel of ''V*'', i.e. the cardinality of the index set ''A'' in the Wold decomposition of ''V''. In other words, a pure isometry of multiplicity ''N'' takes the form
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| :<math>V = \oplus_{1 \le \alpha \le N} S .</math>
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| In this terminology, the Wold decomposition expresses an isometry as a direct sum of a pure isometry and an unitary.
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| A subspace ''M'' is called a [[wandering set|wandering subspace]] of ''V'' if ''V''<sup>''n''</sup>(''M'') ⊥ ''V''<sup>''m''</sup>(''M'') for all ''n'' ≠ ''m''. In particular, each ''M''<sub>''i''</sub> defined above is a wandering subspace of ''V''.
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| == A sequence of isometries ==
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| {{Expand section|date=June 2008}}
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| The decomposition above can be generalized slightly to a sequence of isometries, indexed by the integers.
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| == The C*-algebra generated by an isometry ==
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| Consider an isometry ''V'' ∈ ''L''(''H''). Denote by ''C*''(''V'') the [[C*-algebra]] generated by ''V'', i.e. ''C*''(''V'') is the norm closure of polynomials in ''V'' and ''V*''. The Wold decomposition can be applied to characterize ''C*''(''V'').
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| Let ''C''('''T''') be the continuous functions on the unit circle '''T'''. We recall that the C*-algebra ''C*''(''S'') generated by the unilateral shift ''S'' takes the following form
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| :''C*''(''S'') = {''T''<sub>''f''</sub> + ''K'' | ''T''<sub>''f''</sub> is a [[Toeplitz operator]] with continuous symbol ''f'' ∈ ''C''('''T''') and ''K'' is a [[compact operator on Hilbert space|compact operator]]}. | |
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| In this identification, ''S'' = ''T''<sub>''z''</sub> where ''z'' is the identity function in ''C''('''T'''). The algebra ''C*''(''S'') is called the [[Toeplitz algebra]].
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| '''Theorem (Coburn)''' ''C*''(''V'') is isomorphic to the Toeplitz algebra and ''V'' is the isomorphic image of ''T<sub>z</sub>''.
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| The proof hinges on the connections with ''C''('''T'''), in the description of the Toeplitz algebra and that the spectrum of an unitary operator is contained in the circle '''T'''.
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| The following properties of the Toeplitz algebra will be needed:
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| #<math>T_f + T_g = T_{f+g}.\,</math>
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| #<math> T_f ^* = T_{{\bar f}} .</math>
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| #The semicommutator <math>T_fT_g - T_{fg} \,</math> is compact.
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| The Wold decomposition says that ''V'' is the direct sum of copies of ''T''<sub>''z''</sub> and then some unitary ''U'':
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| :<math>V = (\oplus_{\alpha \in A} T_z) \oplus U.</math>
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| So we invoke the [[continuous functional calculus]] ''f'' → ''f''(''U''), and define
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| :<math>
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| \Phi : C^*(S) \rightarrow C^*(V) \quad \text{by} \quad \Phi(T_f + K) = \oplus_{\alpha \in A} (T_f + K) \oplus f(U).
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| </math>
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| One can now verify Φ is an isomorphism that maps the unilateral shift to ''V'':
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| By property 1 above, Φ is linear. The map Φ is injective because ''T<sub>f</sub>'' is not compact for any non-zero ''f'' ∈ ''C''('''T''') and thus ''T<sub>f</sub>'' + ''K'' = 0 implies ''f'' = 0. Since the range of Φ is a C*-algebra, Φ is surjective by the minimality of ''C*''(''V''). Property 2 and the continuous functional calculus ensure that Φ preserves the *-operation. Finally, the semicommutator property shows that Φ is multiplicative. Therefore the theorem holds.
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| == References ==
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| *L. Coburn, The C*-algebra of an isometry, ''Bull. Amer. Math. Soc.'' '''73''', 1967, 722–726.
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| *T. Constantinescu, ''Schur Parameters, Dilation and Factorization Problems'', Birkhauser Verlag, Vol. 82, 1996.
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| *R.G. Douglas, ''Banach Algebra Techniques in Operator Theory'', Academic Press, 1972.
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| *Marvin Rosenblum and James Rovnyak, ''Hardy Classes and Operator Theory'', Oxford University Press, 1985.
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| [[Category:Operator theory]]
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| [[Category:Invariant subspaces]]
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| [[Category:Functional analysis]]
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| [[Category:C*-algebras]]
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| [[Category:Theorems in functional analysis]]
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| [[de:Shiftoperator#Wold-Zerlegung]]
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Hi there, I am Alyson Pomerleau and I think it seems fairly great when you say it. Invoicing is my profession. Mississippi is exactly where his house is. My spouse doesn't like it the way I do but what I truly like doing is caving but I don't have the time recently.
Also visit my web page ... love psychics (Www.Khuplaza.com)