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Under the assumption that $P$ admits an invariant measure $\\mu$ we prove that if $S$ is eventually compact and the spectrum of its generator is nonempty, then $$\\n P(t)-P(s)\\n_{L^1(E,\\mu)} = 2$$ for all $t,s\\ge 0$ with $t\\not=s$. This result is new even when $E = \\R^n$. We also study the behav"},"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/0509309","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"2005-09-14T13:03:43Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"bc4cd75586e0194e88ff4d2d07f18f27e731b5a933ef8cc0ce8965e98f70ea1f","abstract_canon_sha256":"d07126c3092264c540923bbccae500e5140c6d1d4d69a67f2bf4b9d31afc6609"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:40:22.803602Z","signature_b64":"bJxiqFx+ImZkhdpR8yy0y3c1e3bTcPmtFHS8K6q2ohB4lyl4XetbrQH2cFVZQRveVn59waG6C0LhEnEqLeztBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"638e78c267cab4c8bde3d300d64541bcf5f2215d98eaed49f610afcb3e55c929","last_reissued_at":"2026-07-04T14:40:22.803242Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:40:22.803242Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Norm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups","license":"","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Enrico Priola, Jan van Neerven","submitted_at":"2005-09-14T13:03:43Z","abstract_excerpt":"Let $E$ be a real Banach space. 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