Real and complex operator norms

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 $L_q({\mu }_2)$, $1\le p\le q\le \infty$; 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 $\Vert \Re x\Vert \le \Vert x\Vert$ to $L_{\infty }(\mu )$. The inequality $p\le q$ 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.

13 pages; manuscript, July 2004