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We get the unitary equivalent representations $F_{xx*}(M_{z\\psi(z)}-a)$ on $\\mathcal{L}^{2}(\\sigma(|T+a|),\\mu_{|T+a|})$ for any given $T\\in\\mathcal{B}(\\mathbb{H})$, where $\\psi(z)\\in\\mathcal{L}^{\\infty}(\\sigma(|T+a|),\\mu_{|T+a|})$, $a\\in\\rho(T)$, $F_{xx*}(f(xx^*))=f(x^*x)$, $\\mathcal{B}(\\mathbb{H})$ is the set of all bounded linear operator on complex separable Hilbert space $\\mathbb{H}$. Also, we get that if $z\\psi(z)\\in Fix(F_{xx^*})$, then $T$ has a nontrivial invar"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1611.02981","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-11-07T14:41:15Z","cross_cats_sorted":[],"title_canon_sha256":"5543ed058e3a0248cc6534555057e14cc8a3d99a28980853ff80401bd4ea5ac6","abstract_canon_sha256":"a606344d34d37bed26c4250e95e1ad6bba7daf2014f46333af5b47ed462e445c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:26.998177Z","signature_b64":"mVA0oWwPp7UAnPuMIE2I2gXM7LqhyKPI3f6fhAm7N1THFvkCidN6WOWPEKWxISrTP8xtmsvW6ZPaybLSQ49ZAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70fae75e2596089a79a939d6eb786be9ccba75123e066699194cdab5f9c9607c","last_reissued_at":"2026-05-18T00:28:26.997377Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:26.997377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Noncommutative functional calculate and its application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Lvlin Luo","submitted_at":"2016-11-07T14:41:15Z","abstract_excerpt":"In this paper we construct an unitary operator $F_{xx*}$ such that $(F_{xx^{*}})^2=identity$ and $Fix(F_{xx^*})\\neq\\emptyset$. 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