In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman 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 every 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
V = ( ⨁ α ∈ A S ) ⊕ U {\displaystyle V=\left(\bigoplus _{\alpha \in A}S\right)\oplus U}
for some index set A, where S is the unilateral shift on a Hilbert space Hα, and U is a unitary operator (possible vacuous). The family {Hα} 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 ⊃ V ( H ) ⊃ V 2 ( H ) ⊃ ⋯ = H 0 ⊃ H 1 ⊃ H 2 ⊃ ⋯ , {\displaystyle 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 Hi = Vi(H). If one defines
M i = H i ⊖ H i + 1 = V i ( H ⊖ V ( H ) ) for i ≥ 0 , {\displaystyle M_{i}=H_{i}\ominus H_{i+1}=V^{i}(H\ominus V(H))\quad {\text{for}}\quad i\geq 0\;,}
then
H = ( ⨁ i ≥ 0 M i ) ⊕ ( ⋂ i ≥ 0 H i ) = K 1 ⊕ K 2 . {\displaystyle 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 K1 and K2 are invariant subspaces of V. So V(K2) = K2. In other words, V restricted to K2 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 Mi+1: V "shifts" Mi to Mi+1. Suppose the dimension of each Mi is some cardinal number α. We see that K1 can be written as a direct sum Hilbert spaces
K 1 = ⊕ H α {\displaystyle 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 | K 1 ⊕ V | K 2 = ( ⨁ α ∈ A S ) ⊕ U , {\displaystyle 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 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, ⋂ i ≥ 0 H i = { 0 } . {\textstyle \bigcap _{i\geq 0}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 = ⨁ 1 ≤ α ≤ N S . {\displaystyle V=\bigoplus _{1\leq \alpha \leq 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 of V if Vn(M) ⊥ Vm(M) for all n ≠ m. In particular, each Mi 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.
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