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:

• Level number
• Birkhoff–James orthogonality
• adjoint operator
• singular value decomposition

2010 Mathematics Subject Classifications:

• Primary 47L05
• Secondary 15A18

Acknowledgments

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.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Japan Society for the Promotion of Science (Grant Number 19K14561) and Dr. D. S. Kothari Postdoctoral Fellowship and CSIR MHRD.