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.
The Wold decomposition states that every isometry V takes the form for some index set A, where S is the unilateral shift on a Hilbert space Hα, and U is a unitary operator (possible vacuous).
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 Hi = Vi(H).
We see that K1 can be written as a direct sum Hilbert spaces where each Hα is an invariant subspaces of V and V restricted to each Hα is the unilateral shift S. Therefore which is a Wold decomposition of V. 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.
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: The Wold decomposition says that V is the direct sum of copies of Tz and then some unitary U: So we invoke the continuous functional calculus f → f(U), and define One can now verify Φ is an isomorphism that maps the unilateral shift to V: By property 1 above, Φ is linear.