The Hölder continuity of Löwner’s operator in Euclidean Jordan algebras

Löwner’s operator in Euclidean Jordan algebras, defined via the spectral decomposition of the elements of a scalar function, has been widely used in various optimization problems over Euclidean Jordan algebras. In this note, we shall show that Löwner’s operator in Euclidean Jordan algebras is Hölder continuous if and only if the underlying scalar function is Hölder continuous. Such a property will be useful in designing solution methods for symmetric cone programming and symmetric cone complementarity problem.