Real and complex operator norms.pdf

a r X i v : m a t h / 0 5 1 2 6 0 8 v 1 [ m a t h .F A ] 2 7 D e c 2 0 0 5 Real and Complex Operator Norms Olga Holtz? University of California-Berkeley holtz@Math.Berkeley.EDU Michael Karow Technische Universita?t Berlin karow@math.TU-Berlin.de July 2004 Key words. Complexification, normed complexified space, function space, Lp spaces, lp spaces, linear operator, nonnegative operator, norm, absolute norm, conjugation-invariant norm, shift-invariant norm, monotone norm, norm extension, convex function, integral inequalities. AMS subject classification. 47B37, 47B38, 47B65, 46E30, 47A30, 15A04, 15A60. 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 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 satis- fying ‖?x‖ ≤ ‖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. 1 Introduction By a normed complexified vector space we mean a triple (X,XR, ‖ · ‖X), where ? X and XR are vector spaces over C and R, respectively; ? X is the algebraic complexification of XR, i.e. each x∈X can be uniquely written in the form x = x1 + ix2 with x1, x2∈XR, we set x1 :=?x, x2 :=?x; ?Supported by the DFG center “Mathematics for key technologies” in Berlin, Germany. 1 ? the function ‖ · ‖X : X → [0,∞) is a norm on the complex vector space X, in particular ‖λ x‖X = |λ| ‖x‖X for all λ∈C, x∈X. Let (X,XR, ‖ · ‖X) be a normed complexified space, let (Y, ‖ · ‖Y ) be a normed vector space over C, and A : X → Y be a C-linear oper