Some geometrical properties of the decomposable numerical range

Abstract

For 1≤m≤n, the mth decomposable numerical range of an nxn complex matrix A is defined and denoted by

In this note, we determine the conditions on as essentially Hermitian matrix A(i.e.A is normal with colliner eigenvalues) such that is (i) convex, or (ii) simply connected. For a general A, we obtain a sufficient condition for to be starshaped. These answer some questions raised by Bebiano and the second author on the geometrical properties of the decomposable numerical range.

^∗Research partially supported by an NSF grant and a NATO Collaborative research grant.

^∗Research partially supported by an NSF grant and a NATO Collaborative research grant.

Notes

^∗Research partially supported by an NSF grant and a NATO Collaborative research grant.