{"paper":{"title":"Orthogonal vs. Non-Orthogonal Reducibility of Matrix-Valued Measures","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Erik Koelink, Pablo Rom\\'an","submitted_at":"2015-09-21T08:34:21Z","abstract_excerpt":"A matrix-valued measure $\\Theta$ reduces to measures of smaller size if there exists a constant invertible matrix $M$ such that $M\\Theta M^*$ is block diagonal. Equivalently, the real vector space ${\\mathscr A}$ of all matrices $T$ such that $T\\Theta(X)=\\Theta(X) T^*$ for any Borel set $X$ is non-trivial. If the subspace $A_h$ of self-adjoints elements in the commutant algebra $A$ of $\\Theta$ is non-trivial, then $\\Theta$ is reducible via a unitary matrix. In this paper we prove that ${\\mathscr A}$ is $*$-invariant if and only if $A_h={\\mathscr A}$, i.e., every reduction of $\\Theta$ can be per"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06143","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"}