Real linear operators and numerical ranges

TL;DR

This paper surveys the geometric, spectral, and duality properties of real linear operators between complex Banach spaces, introducing their numerical range and demonstrating its convexity in Hilbert spaces, extending classical results.

Contribution

It introduces the concept of numerical range for real linear operators on Hilbert and Banach spaces and proves its convexity in Hilbert spaces, generalizing classical theorems.

Findings

Numerical range of real linear operators on Hilbert spaces is convex.

Unified treatment of linear, antilinear, and real linear operators.

Extension of classical convexity results to real linear operators.

Abstract

Real linear operators between two complex Banach spaces unify naturally two important classes of linear operators and antilinear operators. We give a survey of basic geometric, spectral and duality properties of real linear operators. The main goal of the paper is to introduce the numerical range of real linear operators on both Hilbert and Banach spaces and to study its properties. In particular, we show that the numerical range of real linear operators on complex Hilbert space is always a convex set (on at least two-dimensional spaces). This generalizes the classical result of Hausdorff and Toeplitz for linear operators.