Let H be a Hilbert space. Given a bounded positive definite operator S on H, and a bounded sequence c = {c k } k∈N of non negative real numbers, the pair (S, c) is frame admissible, if there exists a frame {f k } k∈N on H with frame operator S, such that f k 2 = c k , k ∈ N. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use it to get necessary conditions (and to generalize known sufficient conditions) for a pair (S, c), to be frame admissible.1991 Mathematics Subject Classification. Primary 42C15, Secondary 47A05.

We present results about minimization of convex functionals defined over a finite set of vectors in a finite-dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one.

and IAM-CONICET Dedicated to the memory of "el flaco" L. A. Spinetta. AbstractIn this paper we consider two problems in frame theory. On the one hand, given a set of vectors F we describe the spectral and geometrical structure of optimal completions of F by a finite family of vectors with prescribed norms, where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Benedetto-Fickus' frame potential. On the other hand, given a fixed frame F we describe explicitly the spectral and geometrical structure of optimal frames G that are in duality with F and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.Given m ∈ N we denote by I m = {1, . . . , m} ⊆ N and 1 = 1 m ∈ R m denotes the vector with all its entries equal to 1. For a vector x ∈ R m we denote by x ↓ the rearrangement of x in decreasing order, and R m ↓ = {x ∈ R m : x = x ↓ } the set of ordered vectors.Given H ∼ = C d and K ∼ = C n , we denote by L(H , K) the space of linear operators

Given a finite sequence of vectors F 0 in C d we characterize in a complete and explicit way the optimal completions of F 0 obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequence). Indeed, we construct (in terms of a fast algorithm) a vector -that depends on the eigenvalues of the frame operator of the initial sequence F 0 and the sequence of prescribed norms -that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence F 0 we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem.The well known relation between majorization and tracial inequalities with respect to convex functions allow to describe our results in the following equivalent way: given a finite sequence of vectors F 0 in C d we show that the completions with prescribed norms that minimize the convex potential induced by a strictly convex function are structural minimizers, in the sense that they do not depend on the particular choice of the convex potential.