Annals of Functional Analysis

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

Debmalya Sain

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

extreme contractions operator-norm attainment isometry characterization of Hilbert spaces


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.

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