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arxiv: 2511.12019 · v1 · submitted 2025-11-15 · 🧮 math.FA

On perturbation of Hilbert-Schmidt frames

Jyoti , Lalit Kumar Vashisht This is my paper

Pith reviewed 2026-05-17 22:36 UTC · model grok-4.3

classification 🧮 math.FA
keywords Hilbert-Schmidt framesperturbationframe stabilitysequence replacementHilbert spacesframe operatorfunctional analysisinfinite sequences
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The pith

Replacing some elements in a Hilbert-Schmidt frame keeps the frame property when the changes satisfy explicit size and control limits.

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

The paper establishes explicit criteria under which a sequence obtained by replacing finitely or infinitely many elements of a Hilbert-Schmidt frame remains a Hilbert-Schmidt frame. In the finite case the stability bounds depend quantitatively on the size of each replacement and on how many elements are altered. In the infinite case globally controlled perturbations are shown to be sufficient for the perturbed sequence to retain the frame property. A reader would care because these conditions give concrete ways to modify a frame while still guaranteeing that the frame operator stays bounded and invertible.

Core claim

Under structured modifications that replace finitely or infinitely many frame elements, the perturbed sequence retains the Hilbert-Schmidt frame property whenever the replacements obey explicit quantitative bounds. In the finite setting these bounds scale with perturbation size and the number of altered elements. In the infinite setting it is enough that the perturbations remain globally controlled.

What carries the argument

Structured replacement of frame elements under size and global-control conditions that keep the frame operator invertible with controlled constants.

Load-bearing premise

The perturbations must consist of replacing frame elements and, in the infinite case, must satisfy the global control conditions used in the proofs.

What would settle it

A concrete Hilbert-Schmidt frame together with replacements that obey the stated size and control limits yet make the frame operator unbounded or non-invertible would show the criteria fail.

read the original abstract

In this paper, we study perturbation of Hilbert-Schmidt frames under structured modifications, where the perturbation takes the form of replacing finitely or infinitely many frame elements. We establish explicit criteria under which the perturbed sequence retains the Hilbert-Schmidt frame property. In the finite case, the stability bounds depend quantitatively on the perturbation size and the number of altered elements. For the infinite case, we identify sufficient conditions ensuring stability under globally controlled perturbations. Our study includes illustrative examples demonstrating the applicability of the results.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the stability of Hilbert-Schmidt frames under structured perturbations consisting of the replacement of finitely or infinitely many frame elements. It derives explicit criteria ensuring that the perturbed sequence remains a Hilbert-Schmidt frame, supplies quantitative stability bounds in the finite case that depend on the size of the perturbation and the number of altered elements, and provides sufficient conditions for stability in the infinite case when the perturbations satisfy globally controlled bounds. Illustrative examples are included to demonstrate applicability.

Significance. If the derivations hold, the work adds concrete, quantitative results to the perturbation theory of frames in functional analysis. The explicit bounds for finite replacements and the sufficient conditions for infinite replacements under global control offer practical criteria that could be used to assess robustness when frame elements are modified, complementing existing abstract stability results in the literature on Hilbert-Schmidt operators and frames.

minor comments (3)
  1. Introduction: A brief recall or reference to the precise definition of a Hilbert-Schmidt frame (e.g., via the frame operator belonging to the Hilbert-Schmidt class) would improve accessibility for readers outside the immediate subfield.
  2. Section on finite perturbations: The dependence of the stability bound on the number of altered elements is stated quantitatively, but an explicit comparison of the derived bound with the numerical values in the accompanying example would clarify whether the bound is reasonably sharp.
  3. Infinite-case section: The phrase 'globally controlled perturbations' is used without a numbered display of the precise norm or summability condition; adding an equation label here would facilitate cross-references in later arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript, including the accurate summary of the stability results for Hilbert-Schmidt frames under finite and infinite structured perturbations. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from frame operator properties

full rationale

The paper establishes explicit stability criteria for Hilbert-Schmidt frames under structured perturbations (finite or infinite replacements) by deriving bounds and sufficient conditions directly from the definitions of Hilbert-Schmidt frames and the frame operator. The finite-case quantitative bounds depend on perturbation size and number of altered elements, while the infinite case uses globally controlled conditions; both follow from standard operator inequalities and frame properties without reducing to fitted inputs, self-definitions, or self-citation chains. No load-bearing step equates a result to its own assumption by construction, and the work is self-contained against external frame theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard definitions and properties of Hilbert-Schmidt frames and frame operators from functional analysis; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of Hilbert-Schmidt operators and frame inequalities hold in the underlying Hilbert space.
    Invoked implicitly when defining Hilbert-Schmidt frames and their perturbations.

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