A Hölder type inequality and an interpolation theorem in Euclidean Jordan algebras

Abstract

In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element x we associate the eigenvalue vector λ(x) whose components are the eigenvalues of x written in the decreasing order. For any p ∈ [1,∞], we define the spectral p-norm of x to be the p-norm of λ(x) in Rⁿ. In this paper, we show that ||x ◦ y||1 ≤ ||x|| ||y||q, where x ◦ y denotes the Jordan product of two elements x and y in V and q is the conjugate of p. For a linear transformation on V, we state and prove an interpolation theorem relative to these spectral norms. In addition, we compute/estimate the norms of Lyapunov transformations, quadratic representations, and positive transformations on V.