{"paper":{"title":"Curvature inequalities for operators in the Cowen-Douglas class of a planar domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Md. Ramiz Reza","submitted_at":"2016-04-26T17:15:50Z","abstract_excerpt":"Fix a bounded planar domain $\\Omega.$ If an operator $T,$ in the Cowen-Douglas class $B_1(\\Omega),$ admits the compact set $\\bar{\\Omega}$ as a spectral set, then the curvature inequality $\\mathcal K_T(w) \\leq - 4 \\pi^2 S_\\Omega(w,w)^2,$ where $S_\\Omega$ is the S\\\"{z}ego kernel of the domain $\\Omega,$ is evident. Except when $\\Omega$ is simply connected, the existence of an operator for which $\\mathcal K_T(w) = 4 \\pi^2 S_\\Omega(w,w)^2$ for all $w$ in $\\Omega$ is not known. However, one knows that if $w$ is a fixed but arbitrary point in $\\Omega,$ then there exists a bundle shift of rank $1,$ sa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07758","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"}