{"paper":{"title":"Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.RA"],"primary_cat":"math.CV","authors_text":"Anatolii Grinshpan, Dmitry S. Kaliuzhnyi-Verbovetskyi, Hugo J. Woerdeman, Victor Vinnikov","submitted_at":"2015-01-20T20:30:21Z","abstract_excerpt":"We prove that every matrix-valued rational function $F$, which is regular on the closure of a bounded domain $\\mathcal{D}_\\mathbf{P}$ in $\\mathbb{C}^d$ and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization $$F(z)= D + C\\mathbf{P}(z)_n(I-A\\mathbf{P}(z)_n)^{-1} B. $$ Here $\\mathcal{D}_\\mathbf{P}$ is defined by the inequality $\\|\\mathbf{P}(z)\\|<1$, where $\\mathbf{P}(z)$ is a direct sum of matrix polynomials $\\mathbf{P}_i(z)$ (so that appropriate Archimedean and approximation conditions are satisfied), and $\\mathbf{P}(z)_n=\\bigoplus_{i=1}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.05527","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}