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 any 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 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: where V(H) denotes the range of V. The above defined Hi = Vi(H). 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., 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 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. 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 Hi = {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 In this terminology, the Wold decomposition expresses an isometry as a direct sum of a pure isometry and a unitary operator.