{"paper":{"title":"Infinite-dimensional features of matrices and pseudospectra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Avijit Pal, Dmitry V. Yakubovich","submitted_at":"2016-09-27T09:06:55Z","abstract_excerpt":"Given a Hilbert space operator $T$, the level sets of function $\\Psi_T(z)=\\|(T-z)^{-1}\\|^{-1}$ determine the so-called pseudospectra of $T$. We set $\\Psi_T$ to be zero on the spectrum of $T$. After giving some elementary properties of $\\Psi_T$ (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove that for any operator $T$, there is a sequence $\\{T_n\\}$ of finite matrices such that $\\Psi_{T_n}(z)$ tends to $\\Psi_{T}(z)$ uniformly on $\\C$. In this proof, quasitriangular operators play a special role. This is merely an existence result, we do not"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.08325","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"}