Wold Decomposition - Details

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 H_α, and U is an 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:

where V(H) denotes the range of V. The above defined . If one defines

then

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. an 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 K₁ can be written as a direct sum Hilbert spaces

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

which is a Wold decomposition of V.