{"paper":{"title":"Type and cotype with respect to arbitrary orthonormal systems","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Marius Junge, Stefan Geiss","submitted_at":"1994-01-04T17:10:25Z","abstract_excerpt":"Let $\\on_{k \\in \\nz}$ be an orthonormal system on some $\\sigma$-finite measure space $(\\Om,p)$. We study the notion of cotype with respect to $\\Phi$ for an operator $T$ between two Banach spaces $X$ and $Y$, defined by $\\fco T := \\inf$ $c$ such that \\[ \\Tfmm \\pl \\le \\pl c \\pll \\gmm \\hspace{.7cm}\\mbox{for all}\\hspace{.7cm} (x_k)\\subset X \\pl,\\] where $(g_k)_{k\\in \\nz}$ is a sequence of independent and normalized gaussian variables. It is shown that this $\\Phi$-cotype coincides with the usual notion of cotype $2$ iff \\linebreak $\\fco {I_{\\lin}} \\sim \\sqrt{\\frac{n}{\\log (n+1)}}$ uniformly in $n$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9401205","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"}