{"paper":{"title":"Structures and Numerical Ranges of Power Partial Isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hwa-Long Gau, Pei Yuan Wu","submitted_at":"2013-10-18T08:54:43Z","abstract_excerpt":"We derive a matrix model, under unitary similarity, of an $n$-by-$n$ matrix $A$ such that $A, A^2, \\ldots, A^k$ ($k\\ge 1$) are all partial isometries, which generalizes the known fact that if $A$ is a partial isometry, then it is unitarily similar to a matrix of the form ${\\scriptsize\\left[\\begin{array}{cc} 0 & B 0 & C\\end{array}\\right]}$ with $B^*B+C^*C=I$. Using this model, we show that if $A$ has ascent $k$ and $A, A^2, \\ldots, A^{k-1}$ are partial isometries, then the numerical range $W(A)$ of $A$ is a circular disc centered at the origin if and only if $A$ is unitarily similar to a direct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4952","kind":"arxiv","version":1},"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"}