## 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 *: *L*^{2}(*M*) *→* *X*, *U *: *L*^{2}(*M*) *→ **Y *and a nonnegative, bounded, measurable function *σ *: *M → *[0

*, ∞*) such that

*T *= *Um*_{σ}*V *^{†}*,*

with *m*_{σ}* *: *L*^{2}(*M** *) *→ **L*^{2}(*M** *) defined by *m** _{σ}*(

*f*

*) =*

*σf*for all

*f ∈*

*L*

^{2}(

*M*

*). The expansion*

*T*=

*Um*

_{σ}V

^{†}*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*_{σ}V ^{†}* *can also be compared to the SVE of a compact operator *T *: *X **→ **Y *which has the form

*T *= Σ *σ*_{n}*u*_{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*_{n}*u*_{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*

_{σ}*V*

^{†}*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.