{"paper":{"title":"Frames and operators in Schatten classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hu Bingyang, Kehe Zhu, Le Hai Khoi","submitted_at":"2013-02-11T14:50:42Z","abstract_excerpt":"Let $T$ be a compact operator on a separable Hilbert space $H$. We show that, for $2\\le p<\\infty$, $T$ belongs to the Schatten class $S_p$ if and only if $\\{\\|Tf_n\\|\\}\\in \\ell^p$ for \\emph{every} frame $\\{f_n\\}$ in $H$; and for $0<p\\le2$, $T$ belongs to $S_p$ if and only if $\\{\\|Tf_n\\|\\}\\in\\ell^p$ for \\emph{some} frame $\\{f_n\\}$ in $H$. Similar conditions are also obtained in terms of the sequence $\\{<Tf_n,f_n>\\}$ and the double-indexed sequence $\\{<Tf_n,f_m>\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.2490","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"}