{"paper":{"title":"Linear maps on $M_n(\\mathbb{R})$ preserving Schur stable matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Chandrashekaran Arumugasamy, Sachindranath Jayaraman","submitted_at":"2018-02-15T13:53:38Z","abstract_excerpt":"An $n \\times n$ matrix $A$ with real entries is said to be Schur stable if all the eigenvalues of $A$ are inside the open unit disc. We investigate the structure of linear maps on $M_n(\\mathbb{R})$ that preserve the collection $\\mathcal{S}$ of Schur stable matrices. We prove that if $L$ is a linear map such that $L(\\mathcal{S}) \\subseteq \\mathcal{S}$, then $\\rho(L)$ (the spectral radius of $L$) is at most $1$ and when $L(\\mathcal{S}) = \\mathcal{S}$, we have $\\rho(L) = 1$. In the latter case, the map $L$ preserves the spectral radius function and using this, we characterize such maps on both $M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.05531","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"}