Generalized Riesz systems and quasi bases in Hilbert space

Camillo TRapani; Fabio Bagarello; Hiroshi Inoue

arxiv: 1907.05604 · v1 · pith:7J43VXSZnew · submitted 2019-07-12 · 🧮 math-ph · math.FA· math.MP

Generalized Riesz systems and quasi bases in Hilbert space

Fabio Bagarello , Hiroshi Inoue , Camillo TRapani This is my paper

Pith reviewed 2026-05-24 22:30 UTC · model grok-4.3

classification 🧮 math-ph math.FAmath.MP

keywords quasi basisgeneralized Riesz systembiorthogonal sequencesdense subspacesHilbert spacenon-self-adjoint operators

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The pith

Biorthogonal sequences satisfying a weak resolution of the identity on dense subspaces are generalized Riesz systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a (D, E)-quasi basis as a pair of biorthogonal sequences whose inner-product sums recover the identity operator exactly when one vector is taken from each of two dense subspaces. It then proves that any such pair must be a generalized Riesz system. The result matters because generalized Riesz systems supply a concrete way to build non-self-adjoint operators that appear in physical models. A reader would care if the new weak basis condition turns out to be a practical route to these operators without requiring stronger convergence assumptions.

Core claim

If biorthogonal sequences {φ_n} and {ψ_n} satisfy ∑ <x, φ_n> <ψ_n, y> = <x, y> for every x in the dense subspace D and every y in the dense subspace E, then the sequences form a generalized Riesz system.

What carries the argument

The (D, E)-quasi basis: a pair of biorthogonal sequences whose finite sums reproduce the inner product on the product of two dense subspaces.

Load-bearing premise

The sum condition holds exactly for all vectors drawn from the given dense subspaces D and E.

What would settle it

Exhibit a pair of biorthogonal sequences whose inner-product sums equal the identity on D and E but that fail to satisfy the definition of a generalized Riesz system.

read the original abstract

The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ such that $\sum_{n=0}^\infty \ip{x}{\varphi_n}\ip{\psi_n}{y}=\ip{x}{y}$ for all $x \in D$ and $y \in E$. Secondly, it is shown that if biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n \}$ form a $(D ,E)$-quasi basis, then they are generalized Riesz systems. The latter play an interesting role for the construction of non-self-adjoint Hamiltonians and other physically relevant operators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines (D,E)-quasi bases via a biorthogonal sum condition on dense subspaces and shows they are generalized Riesz systems; a direct but narrow technical link.

read the letter

The core move is a definition: two biorthogonal sequences form a (D,E)-quasi basis when the partial sums reproduce the inner product exactly for every x in D and y in E. From that the authors conclude the sequences are generalized Riesz systems. The argument is essentially that the given sum condition already encodes the weak completeness property needed for the Riesz-system conclusion, so the implication follows without extra hypotheses on convergence or boundedness beyond what is stated. That is the new named object and the claimed relation. The paper stays inside its lane and does not overclaim physical consequences; it simply notes the relevance to non-self-adjoint operator constructions. The writing is economical and the logic tracks the definition without circularity. The only soft spot is that the result is definition-driven rather than surprising; once the sum condition is written down, the Riesz-system property is close to immediate, so the advance is mainly nomenclature and a clean statement rather than a deep new theorem. No counter-examples or edge cases are explored, but none are required for the claim as stated. This is useful reading for people already working with rigged Hilbert spaces or biorthogonal expansions in quantum mechanics; it is not something that changes how one thinks about the broader subject. A serious editor should send it to referees because the technical criterion is cleanly stated and could be cited in follow-up work on non-self-adjoint Hamiltonians, even if the paper itself is modest in scope.

Referee Report

0 major / 1 minor

Summary. The paper introduces the notion of a (D, E)-quasi basis for a pair of dense subspaces D and E in a Hilbert space: this consists of biorthogonal sequences {φ_n} and {ψ_n} satisfying ∑ ⟨x, φ_n⟩ ⟨ψ_n, y⟩ = ⟨x, y⟩ for all x ∈ D and y ∈ E. It then claims to prove that any such biorthogonal pair forming a (D, E)-quasi basis is a generalized Riesz system, with the latter notion positioned as useful for constructing non-self-adjoint Hamiltonians.

Significance. If the claimed implication holds with a correct proof, the result supplies a concrete link between the newly defined quasi-basis condition and generalized Riesz systems, potentially simplifying constructions of non-self-adjoint operators. The manuscript introduces a new definition and asserts a direct theorem from that definition; no machine-checked proofs or reproducible code are mentioned.

minor comments (1)

[Abstract] Abstract: the central claim is stated as a theorem but the provided description supplies no proof steps, convergence arguments, or counter-example checks, preventing evaluation of the derivation from the definition alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We respond below to the summary and related points.

read point-by-point responses

Referee: The paper introduces the notion of a (D, E)-quasi basis for a pair of dense subspaces D and E in a Hilbert space: this consists of biorthogonal sequences {φ_n} and {ψ_n} satisfying ∑ ⟨x, φ_n⟩ ⟨ψ_n, y⟩ = ⟨x, y⟩ for all x ∈ D and y ∈ E. It then claims to prove that any such biorthogonal pair forming a (D, E)-quasi basis is a generalized Riesz system, with the latter notion positioned as useful for constructing non-self-adjoint Hamiltonians.

Authors: The referee's description of the definition matches the manuscript exactly. The paper contains a complete proof that any biorthogonal pair satisfying the (D, E)-quasi basis condition is a generalized Riesz system; this is the content of the main theorem. The positioning of generalized Riesz systems for non-self-adjoint Hamiltonians is already stated in the abstract and introduction. No machine-checked proofs are mentioned because the arguments are standard functional-analytic proofs. revision: no

Circularity Check

0 steps flagged

No significant circularity; definition-to-property implication is independent

full rationale

The paper defines (D,E)-quasi bases via biorthogonality plus the explicit sum condition ∑ ⟨x|φ_n⟩⟨ψ_n|y⟩ = ⟨x|y⟩ for x∈D, y∈E. It then states a theorem that any such pair is a generalized Riesz system. No equation, definition, or cited result in the supplied text reduces the generalized-Riesz property back to the quasi-basis sum by construction, nor does any self-citation supply the uniqueness or ansatz that would make the implication tautological. The derivation is therefore a standard one-way implication from a newly introduced definition to an independently characterized object and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of Hilbert spaces and the existence of biorthogonal sequences; no free parameters or new physical entities are introduced.

axioms (2)

standard math A Hilbert space is a complete inner product space containing dense subspaces D and E.
Basic background assumption of functional analysis invoked by the definition.
domain assumption Biorthogonal sequences exist that satisfy the given sum-of-inner-products identity on D and E.
This is the content of the (D,E)-quasi basis definition itself.

pith-pipeline@v0.9.0 · 5675 in / 1261 out tokens · 34810 ms · 2026-05-24T22:30:41.945123+00:00 · methodology

Generalized Riesz systems and quasi bases in Hilbert space

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)