The q-numerical range of a reducible matrix via a normal operator

The boundary of theq-numerical range of a reducible matrix

“…In [4], the q-numerical range of Að, a þ ibÞ is described in terms of the normal operator T and the following result was obtained. THEOREM 3.1 Suppose that is a positive real number and a 2 þ ðb=Þ 2 > 1.…”

“…The following question is raised in [4], from the viewpoint of Kippenhahn [7], to find the relation between the q-numerical range and the classical numerical range of a matrix. Question 1.1 Let 0 < q < 1 and let the polynomial P represent the boundary of F q ðAÞ.…”

“…This negative result indicates that it is impossible to express the boundary of F q ðAð, a þ ibÞÞ in terms of the numerical range of matrices. On the other hand, the authors of this article in [4] construct explicitly a normal operator on a Hilbert space, and show that the boundary of the q-numerical range of the reducible matrix (1) is determined by the boundary of the q-numerical range of the constructed normal operator. In this study, we provide an algorithm for the boundary of the q-numerical range of Að, a þ ibÞ, and compare the numerical approximation produced in [9].…”