RANK OF THE DERIVATIVE OF THE PROJECTION TO SYMMETRIZED POLYDISC

DOI: 10.18173/2354-1059.2024-0016

Abstract

The projection, also called the symmetrization mapping, from spectralball to symmetrized polydisc is closely related to the spectral Nevanlinna-Pickinterpolation problem. We prove that the rank of the derivative of the projectionfrom the spectral unit ball to the symmetrized polydisc is equal to the degree ofthe minimal polynomial of the matrix at which we take the derivative. Therefore,it explains why the corresponding lifting problem is easier when the matrixbase-point is cyclic since it is a regular point of the symmetrization mapping inthe differential sense.