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 R n . In this paper, we show that ‖ x ∘ y ‖ 1 ≤ ‖ x ‖ p ‖ 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 .