The boundary of the range of a constrained sesquilinear form

Theq-numerical ranges of normal operators

“…Since the last decade, the q-numerical range and Davis-Wielandt shell of an operator or a matrix have attracted a great deal of attention and several results have been obtained (cf [1][2][3][4]). In this article we focus on the q-numerical range and the Davis-Wielandt shell of normal operators.…”

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

“…This observation on the cone (14) helps us below to compute the boundary of F q (A(α)) (for α > 1). Setting s = 1 implies that the lines in (15) pass through the (space) curve…”

“…Proof (i),(ii) We substitute w = u 2 +v 2 +z 2 into the equation G(u, v, w) = 0 of the cone (14). Then we get an implicit expression of the function z = Φ 1 (u, v),…”

“…is the elliptical disk with foci at q a and q b, and with eccentricity q [10,13,14], i.e., the boundary ∂F q (diag{a, b}) coincides with the ellipse…”

“…However, there are no currently known procedures for the construction of the polynomial P, and the explicit equation of the boundary of F q (A) is known only when A is normal or 2 × 2 [4,10,12,13,14]. …”