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Left $m$-invertible and $n$-quasi left $m$-invertible operators share a number of properties. Thus, if $S$ is $n$-quasi left $m$-invertible, then $S^n$ is the perturbation by a nilpotent of the direct sum of a left $m$-invertible with the $0$ operator. In particular, if $T=S^*$ (so that $S$ is $n$-quasi $m$-isomertric) and "},"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":"1812.00221","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-12-01T15:59:12Z","cross_cats_sorted":[],"title_canon_sha256":"fb5c854bea422404665f25a95d6fb4dc1e4040946d0897d978142b9af5c3c8a4","abstract_canon_sha256":"fbfcd8417572c3fc4b18b4eb7818ef13700e2b841ddc23eca847a53b9093bce7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:42.797127Z","signature_b64":"ReX4pcQU3PZfZW4le7Yt/gt5lag41/dOskaJLem3ZWyf5mWnPqB0IoadM8GzyY/hfDbd9QA+hJ47nHpxvkJvCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f99dcdc9a2b8cfc11784bc388da59264a490e586127c98dc3c74401b1cc846f","last_reissued_at":"2026-05-17T23:44:42.796609Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:42.796609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On n-quasi left m-invertible operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"B. P. Duggal","submitted_at":"2018-12-01T15:59:12Z","abstract_excerpt":"A Hilbert space operator $S\\in\\B$ is $n$-quasi left $m$-invertible (resp., left $m$-invertible) by $T\\in\\B$, $m,n \\geq 1$ some integers, if $S^{*n}p(S,T)S^n=0$ (resp., $p(S,T)=0$), where $p(S,T)=\\sum_{j=0}^m{(-1)^{m-j}\\left(\\begin{array}{clcr}m\\\\j\\end{array}\\right)T^jS^j}$. Left $m$-invertible and $n$-quasi left $m$-invertible operators share a number of properties. Thus, if $S$ is $n$-quasi left $m$-invertible, then $S^n$ is the perturbation by a nilpotent of the direct sum of a left $m$-invertible with the $0$ operator. 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