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Let $N(T)$ denote the null space of $T$ and $R(T)$ denote the range of $T$.\n  Recall that $C(T) := D(T) \\cap N(T)^{\\perp}$ is called the {\\it carrier space of} $T$ and the {\\it reduced minimum modulus } $\\gamma(T)$ of $T$ is defined as: $$ \\gamma(T) := \\inf \\{\\|T(x)\\| : x \\in C(T), \\|x\\| = 1 \\} .$$\n  Further, we say that $T$\n  {\\it attains its reduced minimum modulus} if there exists $x_0 \\in C(T) $ such that $\\|x_0\\| = 1$ and $\\|T(x_0)\\| ="},"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":"1704.07534","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-04-25T04:35:26Z","cross_cats_sorted":[],"title_canon_sha256":"f59f802f3c55532edd3914c5079c18d854eab2f6b9895415450e50c86d93b43b","abstract_canon_sha256":"c7f8b5722911d7707a59cc33111626b5006df0e27997a583d5bb531c98f915c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:37.137629Z","signature_b64":"V11EQUu7a3ExoEFNUGstG+VRj2RA2jPU/QFH3Jv9qzVCzQMv0eH4u9xbf97UOu54RYpY67eAh4VfwouSIgujDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27944605d0d4268f7877b418a554bb715b8d48edea52a63e6e56a930f30fa71d","last_reissued_at":"2026-05-18T00:26:37.136948Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:37.136948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Operators that attain the reduced minimum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"G. Ramesh, S. H. Kulkarni","submitted_at":"2017-04-25T04:35:26Z","abstract_excerpt":"Let $H_1, H_2$ be complex Hilbert spaces and $T$ be a densely defined closed linear operator from its domain $D(T)$, a dense subspace of $H_1$, into $H_2$. Let $N(T)$ denote the null space of $T$ and $R(T)$ denote the range of $T$.\n  Recall that $C(T) := D(T) \\cap N(T)^{\\perp}$ is called the {\\it carrier space of} $T$ and the {\\it reduced minimum modulus } $\\gamma(T)$ of $T$ is defined as: $$ \\gamma(T) := \\inf \\{\\|T(x)\\| : x \\in C(T), \\|x\\| = 1 \\} .$$\n  Further, we say that $T$\n  {\\it attains its reduced minimum modulus} if there exists $x_0 \\in C(T) $ such that $\\|x_0\\| = 1$ and $\\|T(x_0)\\| ="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.07534","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1704.07534","created_at":"2026-05-18T00:26:37.137049+00:00"},{"alias_kind":"arxiv_version","alias_value":"1704.07534v2","created_at":"2026-05-18T00:26:37.137049+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.07534","created_at":"2026-05-18T00:26:37.137049+00:00"},{"alias_kind":"pith_short_12","alias_value":"E6KEMBOQ2QTI","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"E6KEMBOQ2QTI66DX","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"E6KEMBOQ","created_at":"2026-05-18T12:31:12.930513+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF","json":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF.json","graph_json":"https://pith.science/api/pith-number/E6KEMBOQ2QTI66DXWQMKKVF3OF/graph.json","events_json":"https://pith.science/api/pith-number/E6KEMBOQ2QTI66DXWQMKKVF3OF/events.json","paper":"https://pith.science/paper/E6KEMBOQ"},"agent_actions":{"view_html":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF","download_json":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF.json","view_paper":"https://pith.science/paper/E6KEMBOQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1704.07534&json=true","fetch_graph":"https://pith.science/api/pith-number/E6KEMBOQ2QTI66DXWQMKKVF3OF/graph.json","fetch_events":"https://pith.science/api/pith-number/E6KEMBOQ2QTI66DXWQMKKVF3OF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF/action/storage_attestation","attest_author":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF/action/author_attestation","sign_citation":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF/action/citation_signature","submit_replication":"https://pith.science/pith/E6KEMBOQ2QTI66DXWQMKKVF3OF/action/replication_record"}},"created_at":"2026-05-18T00:26:37.137049+00:00","updated_at":"2026-05-18T00:26:37.137049+00:00"}