A counterexample to the corona theorem for operators on \(H^ 2(\mathbb D^ n)\). (Q1858323)

Recall the corona problem for \(\Omega \subset \mathbb C^n\): given \(g_1\), \dots, \(g_N \in H^{\infty}(\Omega)\) satisfying \[ | g_1(z)| ^2 + \cdots + | g_N(z)| ^2 \geq \delta^2 > 0 \] for all \(z \in \Omega \), find \(f_1\), \dots, \(f_N \in H^{\infty}(\Omega)\) such that \[ f_1g_1 + \cdots + f_Ng_N = 1. \] In the special case where \(\Omega\) is the unit disc in the complex plane, this question was answered in the affirmative by \textit{L. Carleson} [Ann. Math. 76, 547--559 (1962; Zbl 0112.29702)]. Let \(E\) and \(E'\) be Hilbert spaces and \(H^{\infty} (\mathcal{L}(E',E))\) be the space of bounded holomorphic functions in the unit disk \(\mathbb D\) taking values in \(\mathcal{L}(E',E)\). The operator corona problem is the following: given \(G_1,\dots ,G_N\in H^\infty(\mathcal{L}(E',E))\) such that for some positive constant \(c\) \[ \| G_1(\lambda)^*e\| ^2+\dots+\| G_N(\lambda)^*e\| ^2\geq c \| e\| ^2 \] for all \(\lambda \in \mathbb D\) and \(e\in E\), find \(X_1,\dots ,X_N\in H^\infty(\mathcal{L}(E,E'))\) such that \[ X_1(\lambda)^*G_1(\lambda)^*+\cdots X_N(\lambda)^*G_N(\lambda)^*=I_E \] for all \(\lambda \in \mathbb D\). The authors provide counterexamples in special cases in which both the spaces \(E\) and \(E^{\prime}\) are infinite-dimensional. Earlier work by \textit{S. Treil} [Oper. Theory Adv. Appl. 42, 209--280 (1989; Zbl 0699.47009)] established that the operator corona theorem fails in the setting where both \(E\) and \(E^{\prime}\) are infinite-dimensional. The new contributions in this paper are that \(E = E^{\prime}\) is taken to be \(H^{\infty}(\mathbb D^{n-1})\) where \(n\geq 4\); the authors are able to obtain a counterexample that is based on a sequence of vectors which is both uniformly minimal and an unconditional system. In Treil's example, the sequence of vectors constructed is not an unconditional system. The basic construction is inspired by Treil's example and their refinements rely on connecting earlier work of \textit{C. Horowitz} and \textit{D. M. Oberlin} [Indiana University Math. J. 24, 767--772 (1975; Zbl 0294.32002)] to results of \textit{K. Seip} [Invent. Math. 113, 21--39 (1993; Zbl 0789.30025); Proc. Am. Math. Soc. 117, 213--220 (1993; Zbl 0763.30014)] on weighted Bergman spaces. The counterexamples produced are such that certain subspaces associated with the \(G_i\) are invariant with respect to multiplication by \(z_k\) for all \(k = 1, \dots , n\).