On the generalized numerical range with principal character

Nonconvexity of the Generalized Numerical Range Associated with the Principal Character

“…Proof. By direct computation or by the result in [7]. Conversely, suppose x + iy is of the asserted form.…”

“…Let m = 2 < n, and let H be a subgroup of S m . Suppose A ∈ M n is such that νA is Hermitian for some nonzero ν ∈ C. Then W H (A) is just a line segment in C, and is convex (see [7]).…”

“…There are many generalizations of the classical numerical range, and W H (A) is one of the generalizations involving multilinear algebraic structures introduced in [8]. This generalization has drawn the attention of several authors [2,5,6,7]. Very recently, it has been shown [1] that the mth permanental range is related to quantum systems of bosons (particles carrying positive charges).…”

“…This result leads to many interesting useful consequences in theory and applications. It is known [7] that if (m, n) = (2, 2) then W H (A) is convex. However, it was shown in [5] that there exists a normal matrix A ∈ M n such that the permanental range W Sm (A) is not convex if 2 ≤ m ≤ 3 ≤ n. Moreover, the following conjecture was made.…”

A note on product range of 3-by-3 normal matrices

“…Hence W Π k (A) is not necessary convex in general even when A is normal. However, for the 2-by-2 case, Hu and Tam [6] have shown that W Π 2 (A) is a line segment if and only if A is normal. Recently, the authors [7] of this article provide some sufficient conditions for the convexity of W Π 2 (A) when A is a 3 × 3 normal matrix.…”