Singular Value Decomposition - Bounded Operators On Hilbert Spaces

Bounded Operators On Hilbert Spaces

The factorization can be extended to a bounded operator M on a separable Hilbert space H. Namely, for any bounded operator M, there exist a partial isometry U, a unitary V, a measure space (X, μ), and a non-negative measurable f such that

where is the multiplication by f on L2(X, μ).

This can be shown by mimicking the linear algebraic argument for the matricial case above. VT_f V* is the unique positive square root of M*M, as given by the Borel functional calculus for self adjoint operators. The reason why U need not be unitary is because, unlike the finite dimensional case, given an isometry U₁ with nontrivial kernel, a suitable U₂ may not be found such that

is a unitary operator.

As for matrices, the singular value factorization is equivalent to the polar decomposition for operators: we can simply write

and notice that U V* is still a partial isometry while VT_f V* is positive.