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Given an orthogonal projection $P \\in \\mathcal{B}(\\mathcal{H})$ and an operator $D \\in \\mathcal{B}(\\mathcal{H})$, we may write $D=\\begin{bmatrix} D_1& D_2 D_3 & D_4 \\end{bmatrix}$ relative to the decomposition $\\mathcal{H} = \\mathrm{ran}\\, P \\oplus \\mathrm{ran}\\, (I-P)$. In this paper we study the question: for which non-negative integers $j, k$ can we find a normal operator $D$ and an orthogonal projection $P$ such that $\\mathrm{rank}\\, D_2 ="},"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":"1908.07024","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-19T18:57:16Z","cross_cats_sorted":[],"title_canon_sha256":"10e168b59b5ce6b8cec1df7a86a6734b1d71a6816cad573788c09e8af9306410","abstract_canon_sha256":"dd94ff0ec5ceb33ee417f83b1dc34092d1949e4c4e07c719d4e0b9b8dc10596b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:58:30.215791Z","signature_b64":"ezuIGp6x3D6r8uIIQUiq+mwwgDCRbBDYcDlaOD77I7r5bX4e6+WRB+XCANAeoTbWYE1FdjZJGHgYPkNwLwsmDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5de323c7a0bca4cb26a89f876eb7dddc07043eba8a10ae48ca1ee04baf4b4745","last_reissued_at":"2026-07-04T23:58:30.215355Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:58:30.215355Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normal operators with highly incompatible off-diagonal corners","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Heydar Radjavi, Laurent W. 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