The singular value expansion for compact and non-compact operators

Author

Daniel Crane, Michigan Technological UniversityFollow

Date of Award

2020

Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Administrative Home Department

Department of Mathematical Sciences

Advisor 1

Mark Gockenbach

Committee Member 1

Allan Struthers

Committee Member 2

Cecile Piret

Committee Member 3

Brian Fick

Abstract

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 = Um_σV ^†,

with m_σ : L²(M ) → L²(M ) defined by m_σ(f ) = σf for all f ∈ L²(M ). The expansion T = Um_σV ^† 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 = Um_σV ^† can also be compared to the SVE of a compact operator T : X → Y which has the form

T = Σ σ_nu_n ⊗ v_n

where the above sum may be finite or infinite depending on the rank of T . The set {σ_n} is a sequence of positive real numbers that converge to zero if T has infinite rank. Such σ_n are the singular values of T . The sets {v_n} ⊂ X and {u_n} ⊂ Y are orthonormal sets of vectors that satisfy Tv_n = σ_nu_n for all n. The vectors v_n and u_n are the right and left singular vectors of T, respectively. If the essential range, denoted R_ess(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 ∈ R_ess(σ), the essential preimage of the singleton set s , denoted σ_ess⁻¹( {s} ), is finite, then the bounded operator T = Um_σV ^† 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 (variational approximation), the computed singular values converge at an increased rate; the typical case is that the errors in the singular values are asymptotically equal to the square of the errors in the singular vectors (this statement must be modified if the approximations to the left and right singular vectors converge at different rates). Moreover, in the case of variational approximation, the error in the singular vectors can be compared with the optimal approximation error, with the two being asymptotically equal in the typical case.

Recommended Citation

Crane, Daniel, "The singular value expansion for compact and non-compact operators", Open Access Dissertation, Michigan Technological University, 2020.

https://doi.org/10.37099/mtu.dc.etdr/994