Annals of Functional Analysis

On extreme contractions and the norm attainment set of a bounded linear operator

Debmalya Sain

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper we completely characterize the norm attainment set of a bounded linear operator between Hilbert spaces. In fact, we obtain two different characterizations of the norm attainment set of a bounded linear operator between Hilbert spaces. We further study the extreme contractions on various types of finite-dimensional Banach spaces, namely Euclidean spaces, and strictly convex spaces. In particular, we give an elementary alternative proof of the well-known characterization of extreme contractions on a Euclidean space, which works equally well for both the real and the complex case. As an application of our exploration, we prove that it is possible to characterize real Hilbert spaces among real Banach spaces, in terms of extreme contractions on their 2-dimensional subspaces.

Article information

Source
Ann. Funct. Anal., Volume 10, Number 1 (2019), 135-143.

Dates
Received: 14 May 2018
Accepted: 15 June 2018
First available in Project Euclid: 16 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1547629229

Digital Object Identifier
doi:10.1215/20088752-2018-0014

Mathematical Reviews number (MathSciNet)
MR3899962

Zentralblatt MATH identifier
07045491

Subjects
Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46C15: Characterizations of Hilbert spaces

Keywords
extreme contractions operator-norm attainment isometry characterization of Hilbert spaces

Citation

Sain, Debmalya. On extreme contractions and the norm attainment set of a bounded linear operator. Ann. Funct. Anal. 10 (2019), no. 1, 135--143. doi:10.1215/20088752-2018-0014. https://projecteuclid.org/euclid.afa/1547629229


Export citation

References

  • [1] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935), 169–172.
  • [2] A. Blanco and A.Turnšek, On maps that preserve orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 4, 709–716.
  • [3] R. Grzaślewicz, Extreme contractions on real Hilbert spaces, Math. Ann. 261 (1982), no. 4, 463–466.
  • [4] R. C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291–302.
  • [5] R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), no. 2, 265–292.
  • [6] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325–338.
  • [7] A. Koldobsky, Operators preserving orthogonality are isometries, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 5, 835–837.
  • [8] D. Sain, On the norm attainment set of a bounded linear operator, J. Math. Anal. Appl. 457 (2018), no. 1, 67–76.
  • [9] D. Sain, P. Ghosh, and K. Paul, On symmetry of Birkhoff–James orthogonality of linear operators on finite-dimensional real Banach spaces, Oper. Matrices 11 (2017), no. 4, 1087–1095.
  • [10] D. Sain and K. Paul, Operator norm attainment and inner product spaces, Linear Algebra Appl. 439 (2013), no. 8, 2448–2452.
  • [11] P. Wójcik, On certain basis connected with operator and its applications, J. Math. Anal. Appl. 423 (2015), no. 2, 1320–1329.
  • [12] P. Wójcik, A simple proof of the polar decomposition theorem, Ann. Math. Sil. 31 (2017), no. 1, 165–171.