A uniqueness result concerning Schur ideals (Q2781303)

Let \((\alpha_1,\ldots,\alpha_k)\) be a \(k\)-tuple of distinct points with \(|\alpha_j|<1\). Let \(M^+_k\) denote the closed cone of positive semi-definite \(k\) by \(k\) matrices with complex entries. The Schur product of matrices \(A\) and \(B\) is denoted by \(A\ast B\). The non-empty set \({\mathcal I}\subset M^+_k\) is called a Schur ideal if NEWLINENEWLINENEWLINE(1) \(A\), \(B\in{\mathcal I}\) implies \(A+B\in {\mathcal I}\), NEWLINENEWLINENEWLINE(2) \(A\in {\mathcal I}\), \(P\in M^+_k\) implies \(A\ast P\in{\mathcal I}\). NEWLINENEWLINENEWLINEA Pick body is defined to be the set NEWLINE\[NEWLINE {\mathcal P}(\alpha_1,\ldots,\alpha_k)= \{(w_1,\ldots,w_k)\in{\mathbb C}^k:\exists f\in H^\infty({\mathbb D}), \|f\|_\infty\leq 1, f(\alpha_j)=w_j\}. NEWLINE\]NEWLINE The main result of the paper is the following: NEWLINENEWLINENEWLINETheorem. If \({\mathcal I}\subset M^+_k\) is a closed Schur ideal such that NEWLINE\[NEWLINE \{(w_1,\ldots,w_k)\in{\mathbb C}^k:((1-\overline{w}_iw_j)p_{ij})\geq 0\;\forall (p_{ij})\in{\mathcal I}\}= {\mathcal P}(\alpha_1,\ldots,\alpha_k), NEWLINE\]NEWLINE then the Schur ideal generated by \({1\over 1-\overline{\alpha}_i\alpha_j}\) is equal to \(\mathcal I\).