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In this paper we improve the estimates obtained in \\cite{AP2} for certain special classes of functions.\n  In particular, we improve estimates of Kato \\cite{Ka} and show that $$ \\big\\|\\,|S|-|T|\\,\\big\\|\\le C\\|S-T\\|\\log(2+\\log\\frac{\\|S\\|+\\|T\\|}{\\|S-T\\|}) $$ for every bounded operators $S$ and $T$ on Hilbert space. Here $|S|\\df(S^*S)^{1/2}$. Moreover, we show that this inequality is sharp.\n  We prove in this paper that if $f$ is a nondecreasing continuous function on $\\R$ that vanishes o"},"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":"1104.3553","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-04-18T18:11:36Z","cross_cats_sorted":["math.CA","math.CV","math.SP"],"title_canon_sha256":"5beee7bc4dc59f7379eab47333ec819ba3cadde8ccd06c295aaa404b4b096e7f","abstract_canon_sha256":"0202f51e61c23d8a3eed758fffa358c52f79bd54a28b66fd62394bf5d210be10"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:04.505230Z","signature_b64":"A6VF8mdKx3N7MLieP9OPUmrTgS/H8bQcGQ22bo/2svNLwpiq5s/eg9mdimaTRWETZW5o0DyaVgBkto18urDgBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0bf34e7fc820b540ce9799798f333ff73f569e7cb1222e36dbb57e1d77ac29e0","last_reissued_at":"2026-05-18T04:24:04.504690Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:04.504690Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Estimates of operator moduli of continuity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CV","math.SP"],"primary_cat":"math.FA","authors_text":"Aleksei Aleksandrov, Vladimir Peller","submitted_at":"2011-04-18T18:11:36Z","abstract_excerpt":"In \\cite{AP2} we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in \\cite{AP2} for certain special classes of functions.\n  In particular, we improve estimates of Kato \\cite{Ka} and show that $$ \\big\\|\\,|S|-|T|\\,\\big\\|\\le C\\|S-T\\|\\log(2+\\log\\frac{\\|S\\|+\\|T\\|}{\\|S-T\\|}) $$ for every bounded operators $S$ and $T$ on Hilbert space. Here $|S|\\df(S^*S)^{1/2}$. 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