The q-numerical range and the Davis-Wielandt shell of reducible 3 × 3 matrices

Abstract

For a given q ∈ [0, 1], the q-numerical range of an n × n complex matrix A is defined by* y = q}, and it is closely related with the Davis-Wielandt shell of. In this paper, we investigate systematically the q-numerical range of the 3 × 3 matrixand obtain the equation of its boundary by taking advantage of the special shape of W (A(α), A(α) * A(α)). Furthermore, a parametric representation of ∂F q (A(α)) and the construction of a 4 × 4 matrix B q such that Fq(A(α)) = F1(Bq) are discussed. The q-numerical range of a certain normal operator on an infinite Hilbert space of complex valued (Lebesgue) measurable functions is also considered.

The boundary of theq-numerical range of a reducible matrix

“…A further result in [5] shows that for any a > 0, a 6 ¼ 2 and sufficiently small q, there exists a matrix B 2 M 4 such that F q ðAð1, aÞÞ ¼ FðBÞ. In the case a ¼ 2 or q is rather large, there exists a closed half plane H such that F q ðAð1, aÞÞ \ ðCnHÞ ¼ D \ ðCnHÞ, for some circular disc D, and thus F q ðAð1, aÞÞ \ H ¼ FðBÞ \ H for some matrix B.…”

“…Although there have been a number of interesting papers on the boundary of q-numerical range of a matrix, e.g., [5,8,[10][11][12], applicable algorithms for the construction of the polynomials P j 's and the general explicit equation of the boundary of the q-numerical range of matrices were not developed except normal matrices [12], 2 Â 2 matrices [11] or for a special type of reducible 3 Â 3 matrices [5].…”

“…If ¼ 0, then Að0, a þ ibÞ is normal and the boundary of F q ðAð0, a þ ibÞÞ is characterized by Li [8] and Nakazato [12]. If ¼ 1 and b ¼ 0, the q-numerical range of Að1, aÞ is discussed and its boundary equation is described in [5].…”