A H\"older 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 whose components are the eigenvalues of x written in the decreasing order. For any number p between (and including) one and infinity, we define the spectral p-norm of x to be the p-norm of the corresponding eigenvalue vector in the Euclidean n-space. In this paper, we show that for any two elements x and y, the one-norm of the Jordan product xoy is less than or equal to the product of p-norm of x and q-norm of y, where 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.