The Schur-Horn theorem for operators and frames with prescribed norms and frame operator

J. ANTEZANA, P. MASSEY, M. RUIZ Y D. STOJANOFF

ILLINOIS JOURNAL OF MATHEMATICS

Dept. of Mathematics, Univ. of Illinois

Año: 2007 vol. 51 p. 537 - 560

0019-2082

Let H be a Hilbert space. Given a bounded positive definite operator S on H, and a bounded sequence c = {c_k }_{k in N} of non negative real numbers,  the pair (S, c) is frame admissible, if there exists a frame $ f_k }_{k in N} on H with frame operator S, such that ||f_k||^2 = c_k, k in 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.