On Numerical Radius and Crawford Number Attainment Sets of a Bounded Linear Operator

Debmalya Sain; Arpita Mal; Pintu Bhunia; Kallol Paul

doi:10.68381/jca28007

Abstract

We completely characterize the Crawford number attainment set and the numerical radius attainment set of a bounded linear operator on a Hilbert space. We study the intersection properties of the corresponding attainment sets of numerical radius, Crawford number, norm, minimum norm of a bounded linear operator defined on a normed space. Our study illustrates the similarities and the differences of the extremal properties of a bounded linear operator on a Hilbert space and a general normed space.

Author information

Contact details are reproduced from the original publication and may be historical.

Debmalya Sain

Dept. of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India

saindebmalya@gmail.com

Arpita Mal

Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India

arpitamalju@gmail.com

Pintu Bhunia

Department of Mathematics, Jadavpur University, Kolkata 700032, India

pintubhunia5206@gmail.com

Kallol Paul

Department of Mathematics, Jadavpur University, Kolkata 700032, India

kalloldada@gmail.com

Keywords

Numerical radius
Crawford number
norm attainment set
bounded linear operator

Mathematics Subject Classification

47A12; 46B20

References

K. E. Gustafson, D. K. M. Rao: Numerical Range, the Field of Values of Linear Operators and Matrices, Springer, New York (1997).https://doi.org/10.1007/978-1-4613-8498-4_1 Google Scholar
R. C. James: Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265–292.https://doi.org/10.1090/S0002-9947-1947-0021241-4 Google Scholar
D. Sain: On extreme contractions and the norm attainment set of a bounded linear operator, Ann. Funct. Analysis 10(1) (2019) 135–143.https://doi.org/10.1215/20088752-2018-0014 Google Scholar
D. Sain, K. Paul, K. Mandal: On two extremum problems related to the norm of a bounded linear operator, Operators and Matrices 13(2) (2019) 421–432.https://doi.org/10.7153/oam-2019-13-31 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2021. Rights now held by Banach Press.

[1] K. E. Gustafson, D. K. M. Rao: Numerical Range, the Field of Values of Linear Operators and Matrices, Springer, New York (1997).https://doi.org/10.1007/978-1-4613-8498-4_1 Google Scholar

[2] R. C. James: Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265–292.https://doi.org/10.1090/S0002-9947-1947-0021241-4 Google Scholar

[3] D. Sain: On extreme contractions and the norm attainment set of a bounded linear operator, Ann. Funct. Analysis 10(1) (2019) 135–143.https://doi.org/10.1215/20088752-2018-0014 Google Scholar

[4] D. Sain, K. Paul, K. Mandal: On two extremum problems related to the norm of a bounded linear operator, Operators and Matrices 13(2) (2019) 421–432.https://doi.org/10.7153/oam-2019-13-31 Google Scholar