{"paper":{"title":"On normal operator logarithms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Eduardo Chiumiento","submitted_at":"2013-01-04T18:36:50Z","abstract_excerpt":"Let $X,Y$ be normal bounded operators on a Hilbert space such that $e^X=e^Y$. If the spectra of $X$ and $Y$ are contained in the strip $\\s$ of the complex plane defined by $|\\Im(z)|\\leq \\pi$, we show that $|X|=|Y|$. If $Y$ is only assumed to be bounded, then $|X|Y=Y|X|$. We give a formula for $X-Y$ in terms of spectral projections of $X$ and $Y$ provided that $X,Y$ are normal and $e^X=e^Y$. If $X$ is an unbounded self-adjoint operator, which does not have $(2k+1) \\pi$, $k \\in \\ZZ$, as eigenvalues, and $Y$ is normal with spectrum in $\\s$ satisfying $e^{iX}=e^Y$, then $Y \\in \\{\\, e^{iX} \\, \\}\"$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0797","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"}