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Biorthogonal sequences $\\{ \\phi_{n} \\}$ and $\\{ \\psi_{n} \\}$ in a Hilbert space ${\\cal H}$ are said to be regular if $Span\\; \\{ \\phi_{n} \\}$ and $Span\\; \\{ \\psi_{n} \\}$ are dense in ${\\cal H}$. The first purpose is to show that there exists a non-singular positive self-adjoint operator $T_{\\mbox{$f$}}$ in ${\\cal H}$ defined by an ONB $\\mbox{$f$} \\equiv \\{ f_{n} \\}$ in ${\\cal H}$ such that $\\phi_{n}=T_{\\mbox{$f$}} f_{n}$ and $\\psi_{n}= T_{\\mbox{$f$}}^{-1} f_{n}$, $n=0,1, \\cdots$, and suc"},"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":"1604.01967","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2016-04-07T12:11:19Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"fad626dc30f6d6ebedd568dfdcfe353b17b5c080a241d9052d4816247a0f90ce","abstract_canon_sha256":"bd7f934eafda55feabe138f1c6f5c07aa1a978a82a820d0249344f0ab6f793ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:21.025341Z","signature_b64":"X7pIjulbrIttiw8xRuB/yacGybjEwXByjrmk6IWTj/QFthYReMm4Lee1Jta3FBqEjuvxK7xTbbEroi5O/Cj3CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86c91e84dff46c26af05dcd406c9b4b83856db6ff8473f27cb234b5a8619f772","last_reissued_at":"2026-05-18T01:04:21.024873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:21.024873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"General theory of regular biorthogonal pairs and its physical applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"H. 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