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We prove that if $\\int |M_nf|^r d\\mu \\to 0$, then $f \\in \\overline{(I-T)L^r}$, and show that the converse fails whenever $\\theta$ is ergodic aperiodic. When $\\theta$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r \\in (I-T)L^r$ for which $M_nf_r$ does not converge a.e. (although $\\int |M_nf|^r d\\mu \\to 0$). We further establish t"},"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":"2401.00567","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DS","submitted_at":"2023-12-31T19:05:09Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"b1156c4bb1bd1eef79f959b538d6051c4f481b05ba78bab525f252ffa816ceed","abstract_canon_sha256":"3cc6a5b2ac8befe279ab5fafebeada0569d62439a9684370e431d77bfabea363"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:29:20.263090Z","signature_b64":"y1nvzwccdDt0nUDnuJjTTLveXvP0CyA9VY5RWFN52Y2v0xCkwc7dAruNM4gnvyfCXWEe1G7NC0A4xfpMR7S5CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5ffecb990d2766a92bf05da8f995a2cf738052f4bad417f8ad461bca6ec6ee7","last_reissued_at":"2026-07-05T07:29:20.262616Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:29:20.262616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mean ergodic theorems in $L^r(\\mu)$ and $H^r(\\mathbb T)$, $0<r<1$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.DS","authors_text":"el Houcein el Abdalaoui, Michael Lin","submitted_at":"2023-12-31T19:05:09Z","abstract_excerpt":"Let $T$ be the Koopman operator of a measure preserving transformation $\\theta$ of a probability space $(X,\\Sigma,\\mu)$. We study the convergence properties of the averages $M_nf:=\\frac1n\\sum_{k=0}^{n-1}T^kf$ when $f \\in L^r(\\mu)$, $0<r<1$. We prove that if $\\int |M_nf|^r d\\mu \\to 0$, then $f \\in \\overline{(I-T)L^r}$, and show that the converse fails whenever $\\theta$ is ergodic aperiodic. When $\\theta$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r \\in (I-T)L^r$ for which $M_nf_r$ does not converge a.e. (although $\\int |M_nf|^r d\\mu \\to 0$). 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