{"paper":{"title":"Properties of vector-valued submodules on the bidisk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Constanze Liaw, Kelly Bickel","submitted_at":"2015-06-09T03:06:27Z","abstract_excerpt":"In previous work, the authors studied the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on two-variable model spaces $H^2(\\mathbb{D}^2)\\ominus \\theta H^2(\\mathbb{D}^2)$, where $\\theta$ is a two-variable scalar inner function. Among other results, the authors used Agler decompositions to characterize the ranks of the operators $[S_{z_j}, S^*_{z_j}]$ in terms of the degree of rational $\\theta.$ In this paper, we examine similar questions for $H^2(\\mathbb{D}^2)\\ominus \\Theta H^2(\\mathbb{D}^2)$ when $\\Theta$ is a matrix-valued inner function. We extend several results our previous work connec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02759","kind":"arxiv","version":1},"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"}