Approximating the Singular Value Expansion of a Compact Operator

Crane, Daniel K.; Gockenbach, Mark S.; Roberts, Matthew J.

doi:10.1137/18m1226002

Cited by 2 publications

(3 citation statements)

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“…More precisely, one has is valid in B(H). The difference to the present paper is that, in [3], the expansion is not in eigenvalues and eigenvectors/principal vectors of the compact operator T itself.…”

Section: Comparison Of Present Expansion Results With Known Ones In An Abstract Hilbert Spacementioning

confidence: 74%

“…The most recent publication on expansions of a compact operator in an abstract Hilbert space the author has found is [3]. There, it is used that the singular values and singular vectors of T are related to the nonzero eigenvalues and corresponding eigenvectors of T * T and T T * .…”

Section: Comparison Of Present Expansion Results With Known Ones In An Abstract Hilbert Spacementioning

confidence: 99%

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Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert Space

Kohaupt

2021

Communications in Advanced Mathematical Sciences

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For a symmetric linear compact resp. symmetric densely defined linear operator with compact inverse, expansion theorems in series of eigenvectors are known. The aim of the present paper is to generalize the known expansion theorems to the case of corresponding operators without the symmetry property. For this, we replace the set of orthonormal eigenvectors in the symmetric case by a set of biorthonormal eigenvectors resp. principal vectors in the case of simple eigenvalues resp. general eigenvalues. The results for the operators without the symmetry property are all new. Furthermore, if the operators are symmetric, the generalized results deliver the known expansions. As an application of the results for nonsymmetric operators with simple eigenvalues, we obtain a known expansion in a series of eigenfunctions for a non-selfadjoint Boundary Eigenvalue Problem with ordinary differential operator discussed in a book of Coddington/Levinson. But, we obtain a new result if the eigenvalues are general, that is, not necessarily simple. In addition, for a differential operator of 2nd order with constant coefficients, the eigenfunctions and Green's function are explicitly determined. This result is also new, as far as the author is aware.

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Section: Comparison Of Present Expansion Results With Known Ones In An Abstract Hilbert Spacementioning

confidence: 74%

Section: Comparison Of Present Expansion Results With Known Ones In An Abstract Hilbert Spacementioning

confidence: 99%

Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert Space

Kohaupt

2021

Communications in Advanced Mathematical Sciences

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“…This chapter is an expanded version of our paper [6] that has been accepted for publication in the SIAM Journal on Numerical Analysis (SINUM). Several proofs that were cut out of the paper for economy of space are included here.…”

Section: Chaptermentioning

confidence: 99%

The singular value expansion for compact and non-compact operators

Crane¹

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Approximating the Singular Value Expansion of a Compact Operator

Cited by 2 publications

References 12 publications

Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert Space

Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert Space

The singular value expansion for compact and non-compact operators

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