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

TL;DR

This paper characterizes the norm attainment set of bounded linear operators on Hilbert spaces, explores extreme contractions in finite-dimensional Banach spaces, and provides a new proof of a key characterization for Hilbert spaces.

Contribution

It offers a complete characterization of the norm attainment set, a new elementary proof for extreme contractions on Hilbert spaces, and a method to identify real Hilbert spaces via two-dimensional subspaces.

Findings

Complete characterization of the norm attainment set.

Elementary proof of extreme contractions on Hilbert spaces.

Characterization of real Hilbert spaces using two-dimensional subspaces.

Abstract

In this paper we completely characterize the norm attainment set of a bounded linear operator on a Hilbert space. This partially answers a question raised recently in [\textit{D. Sain, On the norm attainment set of a bounded linear operator, accepted in Journal of Mathematical Analysis and Applications}]. 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 Hilbert space, that 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 two-dimensional subspaces of it.