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In this survey, after having described the above known results, we consider Hardy-Orlicz $H^\\Psi$ and Bergman-Orlicz ${\\mathfrak B}^\\Psi$ spaces, characterize the compactness of their composition operators, and show that there exist Orlicz functions for which there are composition operators which are compact on $H^\\Psi$ but not on ${\\mathfrak B}^\\Psi$."},"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":"1010.6207","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-10-29T13:52:11Z","cross_cats_sorted":[],"title_canon_sha256":"ec5d1da43c8de5efcce9b32fb88b90b4fffebf400a0989f1e7d63e9a2d3630c2","abstract_canon_sha256":"8e7a1a098107cae771bf1104e6451161deb59251bae7f63278832726ac68cdb8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:23.030748Z","signature_b64":"bGNIW0RS1DkWGBuME+f+RArm0aXPMqAoXzkGAwsXGrIYcPlBOJT3HFGCxeMUPgXJpQq8DO6uPFqZv3SYD024Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"53f4efdd5bfa91d75dd316581edc50f1c6752dfeaeba2f1918934e41832ec1c5","last_reissued_at":"2026-05-18T04:26:23.030337Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:23.030337Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Daniel Li (LML)","submitted_at":"2010-10-29T13:52:11Z","abstract_excerpt":"It is known, from results of B. 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