{"paper":{"title":"Submodular spectral functions of principal submatrices of a hermitian matrix, extensions and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.SP","authors_text":"S. Friedland, S. Gaubert","submitted_at":"2010-07-20T18:43:35Z","abstract_excerpt":"We extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions. We show that if $f$ is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix $A$, then the function $I\\mapsto {\\rm tr} f(A[I])$ is supermodular, meaning that ${\\rm tr} f(A[I])+{\\rm tr} f(A[J])\\leq {\\rm tr} f(A[I\\cup J])+{\\rm tr} f(A[I\\cap J])$, where $A[I]$ denotes the $I\\times I$ principal submatrix of $A$. We discuss extensions to self-adjoint operators on infinite dimensional Hilbert spac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.3478","kind":"arxiv","version":4},"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"}