Linear operators preserving the numerical radius of matrices

Abstract

In this note we characterize all the linear operators on the linear space of n × n n \times n complex matrices which preserve the numerical radius and all the linear operators on the real linear space of n × n n \times n Hermitian matrices which preserve the numerical radius. From our results, we easily deduce V. J. Pellegrini’s characterization of all linear operators that preserve the numerical ranges of matrices and the result of Marcus and Moyls concerning the linear operators that preserve the eigenvalues of Hermitian matrices.