A Riesz-Thorin type interpolation theorem in Euclidean Jordan algebras

In a Euclidean Jordan algebra $V$ of rank $n$ which carries the trace inner product, to each element $a$ we associate the eigenvalue vector $\lambda (a)$ in $R^n$ whose components are the eigenvalues of $a$ written in the decreasing order. For any $p\in [1,\mathrm{\infty }]$, we define the spectral $p$-norm of $a$ to be the $p$-norm of $\lambda (a)$ in $R^n$. In a recent paper, based on the $K$-method of real interpolation theory and a majorization technique, we described an interpolation theorem for a linear transformation on $V$ relative to the same spectral norm. In this paper, using standard complex function theory methods, we describe a Riesz-Thorin type interpolation theorem relative to two different spectral norms. We illustrate the result by estimating the norms of certain special linear transformations such as Lyapunov transformations, quadratic representations, and positive transformations.