JOINT RANGES OF HERMITIAN MATRICES AND SIMULTANEOUS DIAGONALIZATION

Let A = (A1,..., A(k)) be a k-tuple of Hermitian operators on an n dimensional inner product space X with unit sphere S = {u member-of X:(u, u) = 1}. With alpha = (alpha-1,..., alpha-k) as the corresponding quadratic forms, i.e., alpha-j(x) = (x, A(j)x) for all x member-of X, j = 1,..., k, we study ranges of alpha over various subsets T of X. The interplay between geometric properties of alpha(T) and algebraic properties of A is our main concern. In particular, we characterize those A for which alpha(S) is identical with the convex hull of the joint spectrum of A defined by sigma(A) = {lambda member-of C(k):A(j)u = lambda(j)u for some u member-of S}. The closely related problem of diagonalizing several Hermitian operators simultaneously is also studied.