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 Lp(μ1) to Lq(μ2), 1 ≤ p ≤ q ≤ ∞; 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 ‖Rx‖ ≤ ‖x‖ to L∞(μ). The inequality p ≤ 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.