# A Riesz-Thorin type interpolation theorem in Euclidean Jordan algebras

## TL;DR

This paper extends interpolation theorems to Euclidean Jordan algebras using complex analysis, providing new bounds for linear transformations with spectral norms.

## Contribution

It introduces a Riesz-Thorin type interpolation theorem in Euclidean Jordan algebras using complex methods, expanding previous real interpolation results.

## Key findings

- Established a Riesz-Thorin type interpolation theorem for spectral norms.
- Applied the theorem to estimate norms of Lyapunov and quadratic transformations.
- Provided bounds for positive linear transformations in Euclidean Jordan algebras.

## Abstract

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.