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