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For example, the concept of shift-like semi-Fredholm operator on Hilbert space coincide with that of upper semi-Browder operator. For applications of Samuel multiplicities we characterize the sets of $\\bigcap_{C\\in B(K,\\,H)}\\sigma_{ab}(M_{C}),\\bigcap_{C\\in\n  B(K,\\,H)}\\sigma_{sb}(M_{C})$ and $\\bigcap_{C\\in\n  B(K,\\,H)}\\sigma_{b}(M_{C}),$ respectively, where $M_{C}=({array}{cc}A&C 0&B {array})"},"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":"1107.0356","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-07-02T04:16:19Z","cross_cats_sorted":[],"title_canon_sha256":"edc7bf989cc657c6090e9ab0f98ba759993b9dfdf76a70e2c5ab04b24e543932","abstract_canon_sha256":"8147a1f0cfa593b80652e470b0498b67d268402b30cd5777778e6b4799ff1a2d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:00.799114Z","signature_b64":"ghnq0DR4KoennNXyJwHHrq0zFCIxi911m6BKOk0DFgc+Ga3l1AnztpsFPYZYJOevVJxZYvkJBGToJjCvxbHRAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"808cb01253177bc24a30437ef00e220be34fe25c661e568d3828f82f33bbcbb6","last_reissued_at":"2026-05-18T00:31:00.798488Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:00.798488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Samuel multiplicities and Browder Spectrum of Operator Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Junde Wu, Shifang Zhang","submitted_at":"2011-07-02T04:16:19Z","abstract_excerpt":"we show that the definitions of some classes of semi-Fredholm operators, which use the language of algebra and first introduced by X. 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