Level numbers of a bounded linear operator between normed linear spaces and singular value decomposition revisited

Debmalya Sain; Saikat Roy; Ryotaro Tanaka

doi:10.1080/03081087.2020.1857677

Level numbers of a bounded linear operator between normed linear spaces and singular value decomposition revisited

Debmalya Sain, Saikat Roy, Ryotaro Tanaka

Liberal Arts

Research output: Contribution to journal › Article › peer-review

Abstract

We introduce the notion of level numbers of a bounded linear operator between normed linear spaces, as a generalization of the singular values of an operator between inner product spaces. We study the geometric and the analytic properties of the level numbers, in connection with Birkhoff–James orthogonality and norm optimization problems. We also illustrate the similarities and the differences between the level numbers and the singular values of an operator. As an application of the present study, we obtain a new and elementary approach to the singular value decomposition of matrices.

Original language	English
Journal	Linear and Multilinear Algebra
DOIs	https://doi.org/10.1080/03081087.2020.1857677
Publication status	Accepted/In press - 2020

Keywords

adjoint operator
Birkhoff–James orthogonality
Level number
singular value decomposition

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title = "Level numbers of a bounded linear operator between normed linear spaces and singular value decomposition revisited",

abstract = "We introduce the notion of level numbers of a bounded linear operator between normed linear spaces, as a generalization of the singular values of an operator between inner product spaces. We study the geometric and the analytic properties of the level numbers, in connection with Birkhoff–James orthogonality and norm optimization problems. We also illustrate the similarities and the differences between the level numbers and the singular values of an operator. As an application of the present study, we obtain a new and elementary approach to the singular value decomposition of matrices.",

keywords = "adjoint operator, Birkhoff–James orthogonality, Level number, singular value decomposition",

author = "Debmalya Sain and Saikat Roy and Ryotaro Tanaka",

note = "Funding Information: The research of Dr. Debmalya Sain is sponsored by Dr. D. S. Kothari Postdoctoral Fellowship under the mentorship of Professor Gadadhar Misra. The research of Mr. Saikat Roy is supported by CSIR MHRD in form of Junior Research Fellowship under the supervision of Prof. Satya Bagchi. The third author was supported in part by Grants-in-Aid for Scientific Research Grant Number 19K14561, Japan Society for the Promotion of Science. ",

year = "2020",

doi = "10.1080/03081087.2020.1857677",

language = "English",

journal = "Linear and Multilinear Algebra",

issn = "0308-1087",

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T1 - Level numbers of a bounded linear operator between normed linear spaces and singular value decomposition revisited

AU - Sain, Debmalya

AU - Roy, Saikat

AU - Tanaka, Ryotaro

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PY - 2020

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N2 - We introduce the notion of level numbers of a bounded linear operator between normed linear spaces, as a generalization of the singular values of an operator between inner product spaces. We study the geometric and the analytic properties of the level numbers, in connection with Birkhoff–James orthogonality and norm optimization problems. We also illustrate the similarities and the differences between the level numbers and the singular values of an operator. As an application of the present study, we obtain a new and elementary approach to the singular value decomposition of matrices.

AB - We introduce the notion of level numbers of a bounded linear operator between normed linear spaces, as a generalization of the singular values of an operator between inner product spaces. We study the geometric and the analytic properties of the level numbers, in connection with Birkhoff–James orthogonality and norm optimization problems. We also illustrate the similarities and the differences between the level numbers and the singular values of an operator. As an application of the present study, we obtain a new and elementary approach to the singular value decomposition of matrices.

KW - adjoint operator

KW - Birkhoff–James orthogonality

KW - Level number

KW - singular value decomposition

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