Real and complex operator norms

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.