On representations of limit relations (Q2784970)

It is shown that the set of systems of polynomial relations representable by bounded operators (on a Hilbert space) with a prescribed maximal norm is closed in a natural topology. In particular, if \(\Gamma_n \subset \mathbb C^{n+1}\) is the set of all \((n+1)\)-tuples \((\lambda_1,\ldots ,\lambda_n,\alpha)\) such that \(\sum_{j=1}^n\lambda_jP_j=\alpha I\) for some orthoprojections \(P_j\), then \(\Gamma_n\) is closed in \(\mathbb C^{n+1}\).