Wold's Decomposition

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

:$V = \left(\bigoplus_ S\right) \oplus U$

for some index set ''A'', where ''S'' is the unilateral shift on a Hilbert space ''H_α'', 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:

:$H = H \supset V(H) \supset V^2 (H) \supset \cdots = H_0 \supset H_1 \supset H_2 \supset \cdots,$

where ''V''(''H'') denotes the range of ''V''. The above defined ''H''_''i'' = ''V''^''i''(''H''). If one defines

:$M_i = H_i \ominus H_ = V^i (H \ominus V(H)) \quad \text \quad i \geq 0 \;,$

then

:$H = \left( \bigoplus_ M_i \right) \oplus \left( \bigcap_ H_i \right) = K_1 \oplus K_2.$

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 ''M_i'' is isomorphic to another, with ''V'' being an isomorphism between ''M_i'' and ''M''_''i''+1: ''V'' "shifts" ''M_i'' to ''M''_''i''+1. Suppose the dimension of each ''M_i'' is some cardinal number ''α''. We see that ''K''₁ can be written as a direct sum Hilbert spaces

:$K_1 = \oplus H_$

where each ''H_α'' is an invariant subspaces of ''V'' and ''V'' restricted to each ''H_α'' is the unilateral shift ''S''. Therefore

:$V = V \vert_ \oplus V\vert_ = \left(\bigoplus_ S \right) \oplus U,$

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, $\bigcap_ H_i = \.$ 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

:$V = \bigoplus_ S .$

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 ''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 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'') = .

In this identification, ''S'' = ''T''_''z'' where ''z'' is the identity function in ''C''(T). The algebra ''C*''(''S'') is called the Toeplitz algebra.

Theorem (Coburn) ''C*''(''V'') is isomorphic to the Toeplitz algebra and ''V'' is the isomorphic image of ''T_z''.

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:

#$T_f + T_g = T_.\,$
#$T_f ^* = T_ .$
#The semicommutator $T_fT_g - T_ \,$ is compact.

The Wold decomposition says that ''V'' is the direct sum of copies of ''T''_''z'' and then some unitary ''U'':

:$V = \left( \bigoplus_ T_z \right) \oplus U.$

So we invoke the continuous functional calculus ''f'' → ''f''(''U''), and define

:$\Phi : C^*(S) \rightarrow C^*(V) \quad \text \quad \Phi(T_f + K) = \bigoplus_ (T_f + K) \oplus f(U).$

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 ''T_f'' is not compact for any non-zero ''f'' ∈ ''C''(T) and thus ''T_f'' + ''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

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{{Hilbert space

Operator theory
Invariant subspaces
C*-algebras
Theorems in functional analysis

de:Shiftoperator#Wold-Zerlegung