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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 $n$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/9401205","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1994-01-04T17:10:25Z","cross_cats_sorted":[],"title_canon_sha256":"bb32c0c13b3c6074ba298b7b1405259f0abad559829c89ceea643685b9eb4008","abstract_canon_sha256":"2c1f943b4128d4fcc2fc33c7dcf8dd8c18731e04131decca5c1ef565e6a7c58c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.734810Z","signature_b64":"4CHojrHsx/BM+WjNgQonrq31HQsgmxVcc+A8DZaRF1WYFLDc4YgOqibL1Hvc7XXoBHFDaKJI3toXzpdbs8BJCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cce2449ed81235c59346ce54c4dbe80a24a022a25658427c159f66676e8e33f","last_reissued_at":"2026-05-18T01:05:51.734345Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.734345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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