  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 $p\in [1,\infty]$, 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.
