Page 119 - Textos de Matemática Vol. 44

CONNECTEDNESS, HESSIANS AND GENERALIZED NUMERICAL RANGES 109
Motivated by Westwick’s paper, Ra¨ıs [24] considered a generalized numerical range associated with a compact Lie group. The following is Ra¨ıs [24] consid- eration. Let K be a compact connected Lie group with Lie algebra k. Let ⟨·, ·⟩ be any AdK-invariant inner product on k, i.e.,
⟨Ad(k)X,Ad(k)Y⟩=⟨X,Y⟩, ∀X,Y ∈k,∀k∈K.
For any X1, X2, C ∈ k, the C-numerical range of the pair (X1, X2) is defined
by
Tam [26] proved that WC(X1,X2) is convex in R2. One may also consider the
joint C-numerical range of X1, . . . , Xp ∈ k defined by
WC(X1,...,Xp) = {(⟨X1,Ad(k)C⟩,...,⟨Xp,Ad(k)C⟩) : k ∈ K}.
Tam’s result is best possible in the sense that WC(X1,...,Xp) fails to be convex in general if p ≥ 3 [5]. The main ideas in Tam’s proof are applying a connectedness result of Atiyah [1] and using the symplectic structure of the co- adjoint orbit. Then the connectedness of the fibres of the map πC : Ad (K )X → R defined by
is established. Here
πC(Y)=⟨C,Y⟩, ∀Y ∈Ad(K)X Ad(K)X := {Ad(k)X : k ∈ K}
WC (X1, X2) = {(⟨X1, Ad (k)C⟩, ⟨X2, Ad (k)C⟩) : k ∈ K}.
is the orbit of X under the adjoint action of K. The convexity of WC(X1,X2) then follows through rotation.
Very recently Markus and Tam [21] gave another proof of the convexity of WC(X1,X2). Without using symplectic technique, they proved the connected- nessofthefibresofthemapfC,X :K→RforallC,X∈kdefinedby
fC,X (k) = ⟨C, Ad (k)X⟩, ∀k ∈ K.
The fibre connectedness result in the compact group K of Markus and Tam is clearly stronger than the fibre connectedness result in the adjoint orbit Ad (K)X:
Ad(·)X
// Ad(K)X πC
zz
since the map Ad (·)X : K → Ad (K )X is continuous.
We shall give a third convexity proof of Tam’s result (see Remark 3.8) via a
connectedness result of Atiyah [1] and a Hessian index result of Duistermaat, Kolk and Varadarajan [10].
K
fC,X
R