The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts

El Hassan Zerouali; Hamza El Azhar; Raul E. Curto; Youssef Omari

arxiv: 2310.13887 · v2 · submitted 2023-10-21 · 🧮 math.FA

The Square Root Problem and Subnormal Aluthge Transforms of Recursively Generated Weighted Shifts

Raul E. Curto , Hamza El Azhar , Youssef Omari , El Hassan Zerouali This is my paper

Pith reviewed 2026-05-24 06:30 UTC · model grok-4.3

classification 🧮 math.FA

keywords weighted shiftssubnormal operatorsAluthge transformMellin transformrecursively generated shiftssquare root problematomic measuresfunctional equation

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The pith

For recursively generated weighted shifts, subnormality and square root problems are equivalent exactly when the associated Mellin transform obeys a functional equation.

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

The paper establishes that the Subnormality Problem and the Square Root Problem for recursively generated unilateral weighted shifts become equivalent if and only if the canonically associated Mellin transform satisfies a specific functional equation. This equivalence is verified directly for all measures with at most six atoms by analyzing the corresponding exponential polynomials. The authors then exhibit an explicit seven-atom measure for which the functional equation fails, producing the first known counterexample to the claimed equivalence. A reader would care because the result settles two open questions from 2009 and 2019 in the classification of subnormal operators and their Aluthge transforms.

Core claim

For recursively generated weighted shifts, (SP) and (SRP) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. For p-atomic measures with p ≤ 6 the equivalence holds, but there exists a 7-atomic measure for which the equivalence fails. This supplies a negative answer to a problem posed by Exner and to a conjecture of Curto et al.

What carries the argument

The canonically associated Mellin transform of the Berger measure, whose functional equation is checked via exponential polynomials on atomic supports and decides equivalence of (SP) and (SRP).

If this is right

The equivalence between (SP) and (SRP) is settled affirmatively for every recursively generated shift whose measure has at most six atoms.
A concrete seven-atomic counterexample demonstrates that the equivalence can fail once the support size reaches seven.
The Mellin-transform criterion supplies a uniform method that replaces earlier case-by-case arguments for low-atomic measures.
The same criterion yields negative answers to the 2009 Exner problem and the 2019 Curto et al. conjecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The appearance of a first failure at seven atoms suggests that the complexity of the functional equation grows with the number of atoms and may produce further counterexamples for larger finite supports.
The Mellin-transform test could be applied to other classes of weighted shifts that admit a recursive structure or an associated transform.
If the functional equation can be rephrased in terms of moment sequences alone, the criterion might extend beyond atomic measures.

Load-bearing premise

The shifts must be recursively generated so that a Mellin transform can be canonically attached and its functional equation examined through exponential polynomials.

What would settle it

An explicit seven-atomic measure together with independent verification that its Mellin transform fails the functional equation while one of (SP) or (SRP) still holds.

read the original abstract

For recursively generated shifts, we provide definitive answers to two outstanding problems in the theory of unilateral weighted shifts: the Subnormality Problem ({\bf SP}) (related to the Aluthge transform) and the Square Root Problem ({\bf SRP}) (which deals with Berger measures of subnormal shifts). We use the Mellin Transform and the theory of exponential polynomials to establish that ({\bf SP}) and ({\bf SRP}) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. For $p$--atomic measures with $p \le 6$, our main result provides a new and simple proof of the above-mentioned equivalence. Subsequently, we obtain an example of a $7$--atomic measure for which the equivalence fails. This provides a negative answer to a problem posed by G.R. Exner in 2009, and to a recent conjecture formulated by R.E. Curto et al in 2019.

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 settles two open problems with an explicit 7-atomic counterexample and a Mellin-transform reduction that works cleanly for p ≤ 6.

read the letter

The main point is that this work gives a negative answer to Exner's 2009 question and the 2019 Curto et al conjecture. For recursively generated weighted shifts, the subnormality problem and the square root problem are equivalent precisely when a certain functional equation holds for the associated Mellin transform. The authors show this equivalence always holds for p-atomic measures when p ≤ 6, then exhibit one 7-atomic measure where it fails. That supplies the first concrete counterexample in the literature. The reduction itself is the other clear contribution: they tie the problems to an identity on exponential polynomials and reduce the check to finite linear algebra on the atomic support. That step looks workable and avoids the circularity that sometimes appears in this area. The stress-test found no inconsistencies in the recurrence setup or the counting argument that separates the p ≤ 6 regime from the p = 7 case, and the construction is described as reproducible from the moment data. One minor soft spot is that the explicit 7-atomic weights and the failure of the functional equation still need direct verification to confirm the measure is positive and the shift is well-defined, but nothing in the stated method suggests a hidden flaw. This is specialized operator-theory work aimed at people already tracking unilateral weighted shifts and subnormal Aluthge transforms. Readers who follow that literature will get immediate use from the counterexample and the functional-equation criterion. The paper deserves a serious referee because it directly resolves the stated open problems with an explicit, checkable construction rather than an existence argument.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that for recursively generated unilateral weighted shifts, the Subnormality Problem (SP) and Square Root Problem (SRP) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. It proves the equivalence for all p-atomic measures with p ≤ 6 via reduction to identities on exponential polynomials and direct linear algebra on the atomic support. It then exhibits an explicit 7-atomic measure where the equivalence fails, furnishing negative answers to a 2009 question of Exner and a 2019 conjecture of Curto et al.

Significance. If correct, the work supplies definitive resolutions to two open problems in weighted-shift theory for the recursively generated class, together with an explicit, reproducible counterexample that separates the p ≤ 6 regime from p = 7. The Mellin-transform criterion and the exponential-polynomial reduction constitute a conceptual advance that may extend beyond atomic measures; the finite linear-algebra verification for the counterexample is a verifiable strength.

minor comments (2)

Abstract: the phrase 'a natural functional equation' is introduced without an immediate pointer to its explicit form or to the section where it is derived; a parenthetical reference to the relevant equation number would improve readability.
The construction of the 7-atomic counterexample is described as reproducible from the recurrence and moment data, but a short appendix tabulating the weights or the support points would make independent verification immediate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation of minor revision. The report correctly captures the main results on the equivalence of SP and SRP for recursively generated shifts via the Mellin transform criterion, the verification for p ≤ 6, and the explicit 7-atomic counterexample.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation associates a Mellin transform canonically to the recursively generated weights, reduces the functional equation to an identity on exponential polynomials, and verifies the identity via direct finite linear algebra on the atomic support. This chain is self-contained and does not reduce to fitted parameters, self-definitional loops, or load-bearing self-citations. The 2019 self-citation merely identifies the conjecture being disproved by the new 7-atomic counterexample and plays no role in establishing the equivalence for p ≤ 6 or the failure at p = 7. The argument relies on external, independently checkable mathematics (Mellin transforms and exponential polynomials) rather than renaming or smuggling prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard properties of the Mellin transform and the theory of exponential polynomials applied to atomic measures; no free parameters or invented entities are introduced.

axioms (1)

standard math Properties of the Mellin transform and exponential polynomials allow checking the functional equation for atomic measures
Invoked to establish equivalence for p≤6 and failure at p=7 (abstract)

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Conjecture C: Mμ(z)Mμ(z+1)=[Mν(z)]² ⇕ ∃ξ with Mμ(z)=[Mξ(z)]²; exponential-polynomial representation via Ritt's theorem

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Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Aluthge, On p-polynomial operators for 0 < p < 1, Integral Equations Op- erator Theory 13(1990), 307–315

A. Aluthge, On p-polynomial operators for 0 < p < 1, Integral Equations Op- erator Theory 13(1990), 307–315

work page 1990
[2]

Ben Taher, M

R. Ben Taher, M. Rachidi and H. Zerouali, On the Aluthge transfo rm of weighted shifts with moments of Fibonacci type. Application to subn ormality, Integral Equations Operator Theory 82(2015), 27–299

work page 2015
[3]

Benhida, R.E

C. Benhida, R.E. Curto and G.R. Exner, Moment inﬁnite divisibility of weighted shifts: sequence conditions, Complex Anal. Oper. Theory 16(2022), art. 5, 1–23

work page 2022
[4]

Conway, The Theory of Subnormal Operators , Mathematical Surveys and Monographs, vol

J.B. Conway, The Theory of Subnormal Operators , Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence (1991)

work page 1991
[5]

Curto and G.R

R.E. Curto and G.R. Exner, Berger measure for some transform ations of subnormal weighted shifts, Integral Equations Operator Theory 84(2016), 49– 450

work page 2016
[6]

Curto and L.A

R.E. Curto and L.A. Fialkow, Recursively generated weighted shift s and the subnormal completion problem, Integral Equations Operator Theory 17(1993), 202–246

work page 1993
[7]

Curto, J

R.E. Curto, J. Kim and J. Yoon, The Aluthge transform of unilater al weighted shifts and the Square Root Problem for ﬁnitely atomic measures, Math. Nachr. 292(2019), 2352–2368

work page 2019
[8]

El Azhar, A

H. El Azhar, A. Hanine, K. Idrissi, and E. H. Zerouali, Square roo t problem and subnormal Aluthge transforms, Ann. Funct. Anal. 14, 8 (2023)

work page 2023
[9]

Exner, Aluthge transforms and n− contractivity of weighted shifts, J

G.R. Exner, Aluthge transforms and n− contractivity of weighted shifts, J. Operator Theory 61(2009), 419–438

work page 2009
[10]

Lee, W.Y

S.H. Lee, W.Y. Lee, and J. Yoon Subnormality of Aluthge transfo rms of weighted shifts, Integral Equations Operator Theory 72(2012), 241–251

work page 2012
[11]

Lee and J

S.H. Lee and J. Yoon, The square root problem and Aluthge transform of weighted shifts , Math. Nachr. 290(2017), 2925–2933

work page 2017
[12]

Ritt, Algebraic combinations of exponentials, Trans

J.F. Ritt, Algebraic combinations of exponentials, Trans. Amer. Math. Soc. 31(1929), 654–679

work page 1929
[13]

Widder, The Laplace Transform, Princeton University Press, 1941

D.V. Widder, The Laplace Transform, Princeton University Press, 1941. THE SQUARE ROOT PROBLEM AND SUBNORMAL ALUTHGE TRANSFORMS 21 R.E. Curto, Department of Mathematics, The University of Io wa, Iowa City, Iowa, U.S.A. Email address : raul-curto@uiowa.edu H. El Azhar, F aculty of sciences, Chouaib Doukkali Universi ty, El Jadida, Morocco. Email address : e...

work page 1941

[1] [1]

Aluthge, On p-polynomial operators for 0 < p < 1, Integral Equations Op- erator Theory 13(1990), 307–315

A. Aluthge, On p-polynomial operators for 0 < p < 1, Integral Equations Op- erator Theory 13(1990), 307–315

work page 1990

[2] [2]

Ben Taher, M

R. Ben Taher, M. Rachidi and H. Zerouali, On the Aluthge transfo rm of weighted shifts with moments of Fibonacci type. Application to subn ormality, Integral Equations Operator Theory 82(2015), 27–299

work page 2015

[3] [3]

Benhida, R.E

C. Benhida, R.E. Curto and G.R. Exner, Moment inﬁnite divisibility of weighted shifts: sequence conditions, Complex Anal. Oper. Theory 16(2022), art. 5, 1–23

work page 2022

[4] [4]

Conway, The Theory of Subnormal Operators , Mathematical Surveys and Monographs, vol

J.B. Conway, The Theory of Subnormal Operators , Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence (1991)

work page 1991

[5] [5]

Curto and G.R

R.E. Curto and G.R. Exner, Berger measure for some transform ations of subnormal weighted shifts, Integral Equations Operator Theory 84(2016), 49– 450

work page 2016

[6] [6]

Curto and L.A

R.E. Curto and L.A. Fialkow, Recursively generated weighted shift s and the subnormal completion problem, Integral Equations Operator Theory 17(1993), 202–246

work page 1993

[7] [7]

Curto, J

R.E. Curto, J. Kim and J. Yoon, The Aluthge transform of unilater al weighted shifts and the Square Root Problem for ﬁnitely atomic measures, Math. Nachr. 292(2019), 2352–2368

work page 2019

[8] [8]

El Azhar, A

H. El Azhar, A. Hanine, K. Idrissi, and E. H. Zerouali, Square roo t problem and subnormal Aluthge transforms, Ann. Funct. Anal. 14, 8 (2023)

work page 2023

[9] [9]

Exner, Aluthge transforms and n− contractivity of weighted shifts, J

G.R. Exner, Aluthge transforms and n− contractivity of weighted shifts, J. Operator Theory 61(2009), 419–438

work page 2009

[10] [10]

Lee, W.Y

S.H. Lee, W.Y. Lee, and J. Yoon Subnormality of Aluthge transfo rms of weighted shifts, Integral Equations Operator Theory 72(2012), 241–251

work page 2012

[11] [11]

Lee and J

S.H. Lee and J. Yoon, The square root problem and Aluthge transform of weighted shifts , Math. Nachr. 290(2017), 2925–2933

work page 2017

[12] [12]

Ritt, Algebraic combinations of exponentials, Trans

J.F. Ritt, Algebraic combinations of exponentials, Trans. Amer. Math. Soc. 31(1929), 654–679

work page 1929

[13] [13]

Widder, The Laplace Transform, Princeton University Press, 1941

D.V. Widder, The Laplace Transform, Princeton University Press, 1941. THE SQUARE ROOT PROBLEM AND SUBNORMAL ALUTHGE TRANSFORMS 21 R.E. Curto, Department of Mathematics, The University of Io wa, Iowa City, Iowa, U.S.A. Email address : raul-curto@uiowa.edu H. El Azhar, F aculty of sciences, Chouaib Doukkali Universi ty, El Jadida, Morocco. Email address : e...

work page 1941