{"paper":{"title":"Compression Covariance and Tangent kernels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"James Tian","submitted_at":"2026-06-10T18:12:57Z","abstract_excerpt":"Let $A\\geq0$ be self-adjoint on a Hilbert space $H$, let $T_{t}=e^{-tA}$, and let $P$ be an orthogonal projection. Relative to the decomposition $H=PH\\oplus P^{\\perp}H$, write \\[ T_{t}=\\begin{pmatrix}C_{t} & V^{*}_{t}\\\\ V_{t} & D_{t} \\end{pmatrix}, \\] where $C_{t}=PT_{t}P|_{PH}$, $V_{t}=P^{\\perp}T_{t}P|_{PH}$, and $D_{t}=P^{\\perp}T_{t}P^{\\perp}|_{P^{\\perp}H}$. The compressed family $\\left(C_{t}\\right)$ consists of positive contractions but need not form a semigroup. Its defect is given by \\[ C_{s+t}-C_{s}C_{t}=V^{*}_{s}V_{t} \\] while the complementary block satisfies \\[ D_{s+t}-D_{s}D_{t}=V_{s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12561","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.12561/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}