Page 14 - Textos de Matemática Vol. 44

4 A. KOVACˇEC
in the style of the Santa Barbara school, i.e. using inner products on the m-fold tensor product of a space, symmetrizers
u1 ⊗···⊗um → 1 uσ−1(1) ⊗···⊗uσ−1(m), etc. m! σ∈Sm
2. Numerical and Determinantal Ranges
Much of Nata´lia’s work is centered on the numerical range and the determi- nantal range. The classical numerical range of a matrix A ∈ Mn = Mn(C) is defined by W(A) = {x∗Ax : x∗x = 1}. According to investigations by Toeplitz and Hausdorff in 1918-19, W(A) is a convex set. As one generalization, for C ∈ Mn(C) one studies the C-numerical range
WC(A) = {tr(AUCU∗) : U ∈ U(n)} = {tr(CUAU∗) : U ∈ U(n)},
where U(n) denotes the group of unitary n×n matrices. If C = diag(γ1, . . . , γn)
this set is often denoted Wc(A) and one has
n
Wc(A) = { γie∗i Aei : (e1, ..., en) an orthonormal basis }.
i=1
If c = (1,0,...,0), then Wc(A) = W(A), i.e. Wc(A) reverts to the classical
numerical range; if c = 1 (1k, 0n−k), with 1k = (1, ..., 1), 0n−k = 0 · 1n−k, this k
is written Wk(A).
Many have worked on numerical ranges before Nat´alia. Much cited in this
respect are papers of Goldberg and Straus in the seventies; in 1975 Westwick
proved via Morse theory that Wc(A) is convex when c ∈ Rn, but if n ≥ 3,
c ∈ Cn it may be not convex, even if A is normal. Poon proved Westwick’s
main result in a more elementary manner in 1980; Tsing showed in 1981 that
Wc(A) is star-shaped with respect to 1 trA γj. Chapter 1 of [HJ] gives an nj
introduction to the numerical range and generalizations and cites much of the literature on these topics till 1990.
The C-determinantal range studied in Nat´alia’s papers often in parallel with the C-numerical range is defined by
∆C(A) = {det(C + UAU∗) : U ∈ U(n)}.
Again, if C = diag(γ1,...,γn) one often writes ∆c(A) for for ∆C(A). Note that WC(A) and ∆C(A) are actually only dependent on the unitary orbit of A. In much of the deeper work on these sets known today one assumes or may assume C to be a diagonal matrix.