The singular value expansion for compact and non-compact operators
Daniel K. Crane · 2020
Given any bounded linear operator T : X → Y between separable Hilbert spaces X and Y , there exists a measure space (M, Α, µ) and isometries V : L2(M) → X, U : L2(M) → Y and a nonnegative, bounded, measurable function σ : M → [0, ∞) such that T = UmσV †, with mσ : L2(M ) → L2(M ) defined by mσ(f ) = σf for all f ∈ L2(M ). The expansion T = UmσV † is called the singular value expansion (SVE) of T . The SVE is a useful tool for analyzing a number of problems such as the computation of the generalized inverse T† of T , understanding the inverse problem Tx = y and, regularizing Tx = y using methods such as Tikhonov regularization. In fact, many standard Tikhonov regularization results can be derived by making use of the SVE. The expansion T = UmσV † can also be compared to the SVE of a compact operator T : X → Y which has the form T = Σ σnun ⊗ vn where the above sum may be finite or infinite depending on the rank of T . The set {σn} is a sequence of positive real numbers that converge to zero if T has infinite rank. Such σn are the singular values of T . The sets {vn} ⊂ X and {un} ⊂ Y are orthonormal sets of vectors that satisfy Tvn = σnun for all n. The vectors vn and un are the right and left singular vectors of T, respectively. If the essential range, denoted Ress(σ), forms a sequence of positive real numbers converging to zero (or is merely a finite set of nonnegative real numbers) and for each nonzero s ∈ Ress(σ), the essential preimage of the singleton set s , denoted σess−1( {s} ), is finite, then the bounded operator T = UmσV † is in fact compact. The converse of this statement is also true. If the operator T is compact, the singular values and vectors of T may be approximated by discretizing the operator and finding the singular value decomposition of a scaled Galerkin matrix. In general, the approximated singular values and vectors converge at the same rate, which is governed by the error (in the operator norm) in approximating T by the discretized operator. However, when the discretization is accomplished by projection