{"paper":{"title":"The Hadamard product and the free convolutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Arijit Chakrabarty","submitted_at":"2015-11-12T04:28:46Z","abstract_excerpt":"It is shown that if a probability measure $\\nu$ is supported on a closed subset of $(0,\\infty)$, that is, its support is bounded away from zero, then the free multiplicative convolution of $\\nu$ and the semicircle law is absolutely continuous with respect to the Lebesgue measure. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is proved and subsequently used. As a byproduct, a result, showing that the limiting spectral distribution of the Hadamard product is same as that of a symmetric random matrix with entries from a mean zero stat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.03770","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"}