Real and complex operator norms

TL;DR

This paper investigates the relationship between real and complex operator norms, providing bounds, identifying classes where they coincide, and constructing norm-preserving extensions from real to complex spaces.

Contribution

It offers new bounds on the ratio of real to complex norms and characterizes classes of operators where these norms are equal, along with constructing norm-preserving extensions.

Findings

Real and complex norms coincide for specific classes of operators.

Bounds on the ratio of real to complex norms are established.

Sharpness of the inequality p ≤ q in certain operator classes.

Abstract

Real and complex norms of a linear operator acting on a normed complexified space are considered. Bounds on the ratio of these norms are given. The real and complex norms are shown to coincide for four classes of operators: 1) real linear operators from $L_p(\mu_1)$ to $L_q(\mu_2)$, $1\leq p\leq q\leq \infty$; 2) real linear operators between inner product spaces; 3) nonnegative linear operators acting between complexified function spaces with absolute and monotonic norms; 4) real linear operators from a complexified function space with a norm satisfying $\|\Re x \|\leq \|x\|$ to $L_\infty(\mu)$. The inequality $p\leq q$ in Case 1 is shown to be sharp. A class of norm extensions from a real vector space to its complexification is constructed that preserve operator norms.