# Wold's decomposition > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Wold's_decomposition > Markdown URL: https://mediated.wiki/source/Wold's_decomposition.md > Source: https://en.wikipedia.org/wiki/Wold's_decomposition > Source revision: 1250363555 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) This article is about the general mathematical result. For the application to time series analysis, see [Wold's theorem](/source/Wold's_theorem). In [mathematics](/source/Mathematics), particularly in [operator theory](/source/Operator_theory), **Wold decomposition** or **Wold–von Neumann decomposition**, named after [Herman Wold](/source/Herman_Wold) and [John von Neumann](/source/John_von_Neumann), is a [classification theorem](/source/Classification_theorem) for [isometric linear operators](/source/Isometry) on a given [Hilbert space](/source/Hilbert_space). It states that every isometry is a direct sum of copies of the [unilateral shift](/source/Unilateral_shift) and a [unitary operator](/source/Unitary_operator). In [time series analysis](/source/Time_series_analysis), the theorem implies that every [stationary](/source/Stationary_process) discrete-time [stochastic process](/source/Stochastic_process) can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a [moving average process](/source/Moving_average_process). ## Details Let *H* be a [Hilbert space](/source/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 - V = \left(\bigoplus_{\alpha \in A} S\right) \oplus U for some index set *A*, where *S* is the [unilateral shift](/source/Unilateral_shift) on a Hilbert space *Hα*, and *U* is a unitary operator (possible vacuous). The family {*H**α*} 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: - H = H \supset V(H) \supset V^2 (H) \supset \cdots = H_0 \supset H_1 \supset H_2 \supset \cdots, where *V*(*H*) denotes the range of *V*. The above defined *H**i* = *V**i*(*H*). If one defines - M_i = H_i \ominus H_{i+1} = V^i (H \ominus V(H)) \quad \text{for} \quad i \geq 0 \;, then - H = \left( \bigoplus_{i \geq 0} M_i \right) \oplus \left( \bigcap_{i \geq 0} H_i \right) = K_1 \oplus K_2. 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](/source/Cardinal_number) *α*. We see that *K*1 can be written as a direct sum Hilbert spaces - K_1 = \oplus H_{\alpha} where each *Hα* is an invariant subspaces of *V* and *V* restricted to each *Hα* is the unilateral shift *S*. Therefore - V = V \vert_{K_1} \oplus V\vert_{K_2} = \left(\bigoplus_{\alpha \in A} S \right) \oplus U, which is a Wold decomposition of *V*. ### Remarks It is immediate from the Wold decomposition that the [spectrum](/source/Spectrum_(functional_analysis)) of any proper, i.e. non-unitary, isometry is the unit disk in the complex plane. An isometry *V* is said to be **pure** if, in the notation of the above proof, \bigcap_{i\ge0} H_i = \{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 - V = \bigoplus_{1 \le \alpha \le N} S . In this terminology, the Wold decomposition expresses an isometry as a direct sum of a pure isometry and a unitary operator. A subspace *M* is called a [wandering subspace](/source/Wandering_set) of *V* if *V**n*(*M*) ⊥ *V**m*(*M*) for all *n* ≠ *m*. In particular, each *M**i* defined above is a wandering subspace of *V*. ## A sequence of isometries The decomposition above can be generalized slightly to a sequence of isometries, indexed by the integers. ## The C*-algebra generated by an isometry Consider an isometry *V* ∈ *L*(*H*). Denote by *C**(*V*) the [C*-algebra](/source/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*). 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 - *C**(*S*) = {*T**f* + *K* | *T**f* is a [Toeplitz operator](/source/Toeplitz_operator) with continuous symbol *f* ∈ *C*(**T**) and *K* is a [compact operator](/source/Compact_operator_on_Hilbert_space)}. In this identification, *S* = *T**z* where *z* is the identity function in *C*(**T**). The algebra *C**(*S*) is called the [Toeplitz algebra](/source/Toeplitz_algebra). **Theorem (Coburn)** *C**(*V*) is isomorphic to the Toeplitz algebra and *V* is the isomorphic image of *Tz*. The proof hinges on the connections with *C*(**T**), in the description of the Toeplitz algebra and that the spectrum of a unitary operator is contained in the circle **T**. The following properties of the Toeplitz algebra will be needed: 1. T_f + T_g = T_{f+g}.\, 1. T_f ^* = T_{{\bar f}} . 1. The semicommutator T_fT_g - T_{fg} \, is compact. The Wold decomposition says that *V* is the direct sum of copies of *T**z* and then some unitary *U*: - V = \left( \bigoplus_{\alpha \in A} T_z \right) \oplus U. So we invoke the [continuous functional calculus](/source/Continuous_functional_calculus) *f* → *f*(*U*), and define - \Phi : C^*(S) \rightarrow C^*(V) \quad \text{by} \quad \Phi(T_f + K) = \bigoplus_{\alpha \in A} (T_f + K) \oplus f(U). One can now verify Φ is an isomorphism that maps the unilateral shift to *V*: By property 1 above, Φ is linear. The map Φ is injective because *Tf* is not compact for any non-zero *f* ∈ *C*(**T**) and thus *Tf* + *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. ## References - Coburn, L. (1967). "The C*-algebra of an isometry". *[Bull. Amer. Math. Soc.](/source/Bulletin_of_the_American_Mathematical_Society)*. **73** (5): 722–726. [doi:10.1090/S0002-9904-1967-11845-7](https://doi.org/10.1090/S0002-9904-1967-11845-7) - Constantinescu, T. (1996). [*Schur Parameters, Factorization and Dilation Problems*](https://books.google.com/books?id=aEwoy7k8ufcC). Vol. 82. Operator Theory, Advances and Applications. Birkhäuser. ISBN 3-7643-5285-X. - Douglas, R. G. (1972). [*Banach Algebra Techniques in Operator Theory*](https://books.google.com/books?id=_xvDk-mOiHIC). Academic Press. ISBN 0-12-221350-5. - Rosenblum, Marvin & Rovnyak, James (1985). [*Hardy Classes and Operator Theory*](https://books.google.com/books?id=PchPAQAAIAAJ). Oxford University Press. ISBN 0-19-503591-7. --- Adapted from the Wikipedia article [Wold's decomposition](https://en.wikipedia.org/wiki/Wold's_decomposition) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Wold's_decomposition?action=history)). 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