Page 117 - Textos de Matemática Vol. 44

CONNECTEDNESS, HESSIANS AND GENERALIZED NUMERICAL RANGES
XUHUA LIU AND TIN-YAU TAM
Dedicated to Nat´alia Bebiano
Abstract. A brief survey on some generalized numerical ranges associated with semisimple Lie algebras is given. We give another proof of the convexity of generalized numerical ranges associated with compact Lie groups via a con- nectedness result of Atiyah and a Hessian index result of Duistermaat, Kolk and Varadarajan.
1. Numerical range and compact Lie group
Let Cn (resp., Rn) be the vector space of all n-tuple complex (resp., real) numbers. Let Cn×n denote the set of all complex n×n matrices. The (classical) numerical range of A ∈ Cn×n is the set
W(A) = {x∗Ax : x ∈ Cn,x∗x = 1} ⊂ C,
which is the image of the unit sphere in Cn under the quadratic map x → x∗Ax. Toeplitz-Hausdorff theorem [27, 14] asserts that W(A) is convex for all A ∈ Cn×n, which perhaps is the most interesting property of numerical range. See [8] for an interesting geometric proof.
There are many generalizations of the classical numerical range motivated by theories and applications in the last decades [12, 18]. Halmos [13] introduced the notion of k-numerical range of A ∈ Cn×n for 1 ≤ k ≤ n, which is defined by
k
Wk(A) = x∗i Axi : x1,...,xk are orthonormal in Cn . i=1
He conjectured and Berger [2] proved that Wk(A) is always convex. West- wick [29] further generalized the k-numerical range to the c-numerical range of
2010 Mathematics Subject Classification. 15A60, 17B20, 47L07
Key words and phrases. Semisimple Lie algebra, generalized numerical range, convexity, ad- joint orbit
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