## Annals of Functional Analysis

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

Debmalya Sain

#### 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
Accepted: 15 June 2018
First available in Project Euclid: 16 January 2019

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

#### 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

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