Optimal frame completions with prescribed norms

MASSEY, PEDRO; RUIZ, MARIANO; STOJANOFF, DEMETRIO

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos (CLAM); 2012

Unión Matemática de América Latina y el Caribe (UMALCA)

Let $mathcal H$ be a $d$-dimensional complex Hilbert space, let $mathcal F_0={f_i}_{i=1}^{n_0}$ be a finite sequence of vectors in $mathcal H$ and let $mathbf a=(a_i)_{i=n_0+1}^n$ be a sequence of positive numbers for some $ngeq max{d,n_0+1}$. We say that a finite sequence $mathcal F={f_i}_{i=1}^n$ of vectors in $mathcal H$ is an $mathbf a$-completion of $mathcal F_0$ if $|f_i|^2 =a_i$, $n_0+1leq ileq n$. In this context, it is desired to construct $mathbf a$-completions $mathcal F={f_i}_{i=1}^n$ that minimize a convex functional of the form $P_g(mathcal F)= ext{tr}(g(S_{mathcal F}))$, where $S_{mathcal F}$ denotes the operator $sum_{i=1}^n f_iotimes f_i$ and $g:[0,infty) ightarrow [0,infty)$ is a certain strictly convex function (e.g. if $g(x)=x^2$ then $P_g(mathcal F)$ is known as the Benedetto-Fickus´ frame potential of $mathcal F$). It turns out that this problem can be studied in terms of the structure of the set $$U_t(S_{mathcal F_0},k)={S_{mathcal F_0}+B: Bin mathcal B(mathcal H)^+, , ext{rank}(B)leq k, , ext{tr}(B)=t} , $$ where $mathcal B(mathcal H)^+$ denotes the cone of positive operators, $k=n-n_0$ and $t=sum_{i=n_0+1}^na_i$. Indeed, as a consequence of the structure of $U_t(S_{mathcal F_0},k)$ we will show that, under certain hypothesis, there exist frames $mathcal F$ that are $mathbf a$-completions of $mathcal F_0$ and that are optimal with respect to the convex functional $P_g$ for every strictly convex function $g$.