{"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"}