Nonconvexity of the permanental numerical range

Nonconvexity of the Generalized Numerical Range Associated with the Principal Character

“…Besides extending the result in [5], our proofs may lead to useful ideas for studying their conjecture and other types of generalized numerical ranges. Furthermore, in view of our results, one may consider strengthening Conjecture 1.1 by removing the condition H = S m .…”

“…The authors of [5] commented that the methods in their paper are too special to be used to deal with the conjecture, and urged for more general techniques. In this note, we make a move along this direction.…”

“…We remark that one can extend the class of normal matrices A such that W H (A) is not convex to a wider class of nonnormal matrices by a simple continuity argument, as in [5,Theorem 4].…”

“…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).…”

“…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.…”