Numerical Ranges Arising from Simple Lie Algebras

Abstract

Abstract. A unified formulation is given to various generalizations of the classical numerical range including the c-numerical range, congruence numerical range, q-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized.

Convexity and star-shapedness of real linear images of special orthogonal orbits

“…However if the codomain of L is R 2 then L(O(A)) is always convex. This result was obtained by Li and Tam [7] by using techniques in Lie algebra. In the following, we shall give an alternative proof on this result by showing that L(O(A)) has convex boundary for all A ∈ R n×n and linear L : R n×n → R 2 , i.e., the intersection of L(O(A)) with any of its supporting lines is path connected.…”

H-joint numerical ranges

“…In studies of the numerical range and its generalisations, a useful technique is reducing the problems to the 2 x 2 case. For instance, convexity results are proven using such a reduction (in clever ways), see [1,5,9,13]. So, in Section 2 we give a complete description of the 2 x 2 case in our study of if-joint numerical ranges, and use this description in Section 3, as well as known results on convexity of joint numerical ranges of Hermitian matrices, to derive convexity results for i/-joint numerical ranges.…”

A range associated with skew symmetric matrix

“…It is a compact convex set in C given by Tam [21,Corollary 2.4 (2), (3)] (with C ¼ 1 [15,20]). When n ¼ 2, i.e.…”