Majorization, Schur-Horn’s theorem and frames

ANTEZANA, JORGE; MASSEY, PEDRO; RUIZ, MARIANO; STOJANOFF, DEMETRIO

CABA

Congreso; 1era. Escuela sobre Análisis Funcional y Geometría; 2011

Instituto Argentino de Matematica "Alberto P. Calderón" - CONICET

Majorization - for real vectors and selfadjoint matrices - is an important tool in matrix analysis. Indeed, majorization is a preorder relation that implies an interesting family of inequalities (e.g. tracial inequalities, inequalities with respect to unitarily invariant norms). The Schur-Horn theorem establishes the following characterization of majorization: given two real vectors x,y in R^n then x is majorized by y if and only if there exists a (complex) n times n selfadjoint matrix A with characteristic polynomial prod_{i=1}^n(x-y_i) and main diagonal x. On the other hand, a frame for a (separable complex) Hilbert space H is a family of vectors F={f_i}_{i in I} in H that allow linear encoding-decoding schemes that areredundant and stable. In case dim H=n is finite and I is a finite set then, a family of vectors F={f_i}_{i in I} in H is a frame if F linearly spans H or equivalently, if the n times n matrix S_F=sum_{i in I} f_i  f_i^* is positive and invertible. There are two problems related with finite frames that have been extensively considered. On the one hand, given a positive invertible n times n matrix S and a finite sequence of positive numbers (a_i)_{i in I}, then one problem is to determine when there exists a frame  F={f_i}_{i in I} such that S_F=S and such that |f_i|^2=a_i, for i in I. The other problem is to determine the structure (both geometrical and spectral) of the frames F={f_i}_{i in I} that minimize the functional P(F)=sum_{i,j in I} |<f_i , f_j >|^2  (known as the frame potential) subject to some (normalization) conditions. In this talk I will show that these and some other problems in frame theory are related with the Schur-Horn´s theorem and majorization between real vectors. I will mention some cases in which the solution can be described in terms of majorization which implies some structural properties of these solutions.  This talk is based on joint work with J. Antezana, M. Ruiz and D. Stojanoff.