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When the commutative condition is weaken to $ab=\\lambda ba ~(\\lambda \\neq 0)$, we also show that $a-b$ is pseudo Drazin invertible if and only if $aa^\\ddag(a-b)bb^\\ddag$ is pseudo Drazin invertible."},"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":"1307.7232","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-07-27T08:08:14Z","cross_cats_sorted":[],"title_canon_sha256":"b35f71c0cadeb31dfb1c7c4d8a6f55201daf3323cb7f5a8bebd30d64431eca73","abstract_canon_sha256":"d3725b4400791920579734e61ec6f59f4ae5563cc63947a78c9a39276b27bd20"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:23.814573Z","signature_b64":"kLuUhs4z1G7iQ9t3imVO9S21oW5yE5+IPzp2zevglrj9Ffd5dkPGDBJWkZaV5Zt1H5FZRcEGn3bY/OKPOPcsBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0cc60d54086084dec46137fbc5797d47e07693105a0de4ecb382edb200354049","last_reissued_at":"2026-05-18T03:17:23.814188Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:23.814188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Additive property of pseudo Drazin inverse of elements in a Banach algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Huihui Zhu, Jianlong Chen","submitted_at":"2013-07-27T08:08:14Z","abstract_excerpt":"We study properties of pseudo Drazin inverse in a Banach algebra with unity 1. 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