The singular value expansion for compact and non-compact operators

Crane, Daniel K.

doi:10.37099/mtu.dc.etdr/994

Cited by 2 publications

(1 citation statement)

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“…Fack and Kosaki [12] defined generalized s-numbers for certain operators in a von Neumann algebra, and their techniques allow for a nonempty continuous spectrum. In the context of the SVE presented in this paper, it would be natural to define the set of s-numbers of a bounded linear operator T : X → Y as the essential range of σ (where T = Um σ V † ); we refer the reader to the first author's PhD dissertation [13] for a discussion. The relationship between these two approaches remains to be investigated.…”

Section: S-numbersmentioning

confidence: 99%

The Singular Value Expansion for Arbitrary Bounded Linear Operators

Crane

Gockenbach

2020

Mathematics

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The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed.

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Section: S-numbersmentioning

confidence: 99%