Operator versions of H\"older inequality and Hilbert $C^*$-modules

Dragoljub J. Ke\v{c}ki\'c

arxiv: 1907.00369 · v1 · pith:GMHEVWMPnew · submitted 2019-06-30 · 🧮 math.FA

Operator versions of H\"older inequality and Hilbert C^*-modules

Dragoljub J. Ke\v{c}ki\'c This is my paper

Pith reviewed 2026-05-25 12:30 UTC · model grok-4.3

classification 🧮 math.FA

keywords Hilbert C*-modulesweighted Cauchy-Schwarz inequalityHölder inequalityunitarily invariant normsoperator theoryC*-algebrasfunctional analysis

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The pith

The weighted Cauchy-Schwarz inequality for Hilbert C*-modules implies multiple Hölder-type inequalities for unitarily invariant norms on Hilbert space operators.

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

This paper shows how a recently established weighted Cauchy-Schwarz inequality in the Hilbert C*-module setting directly produces Hölder-type inequalities. The resulting inequalities apply to unitarily invariant norms on operators between Hilbert spaces. A reader would care because these norms appear throughout operator theory and matrix analysis, and the module approach supplies them from a single algebraic source. The derivation relies on viewing the norms through the module structure rather than through separate analytic arguments. No extra restrictions on the module or the norm are required for the implications to hold.

Core claim

The weighted Cauchy-Schwarz inequality for Hilbert C*-modules leads to many Hölder type inequalities for unitarily invariant norms on Hilbert space operators. The central claim is that this implication follows by direct application of the module inequality to the algebraic structure that underlies the unitarily invariant norms.

What carries the argument

Weighted Cauchy-Schwarz inequality for Hilbert C*-modules, applied to the module structure underlying unitarily invariant norms.

If this is right

Hölder inequalities are obtained for the full class of unitarily invariant norms rather than for isolated cases.
The same module inequality generates multiple distinct Hölder forms by varying the choice of weights and module elements.
Known inequalities for Schatten norms appear as immediate special cases of the general result.

Where Pith is reading between the lines

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

The same direct-application technique could be tried on other module inequalities to produce further operator-norm results.
The approach may connect to existing work on noncommutative Hölder inequalities without requiring new module constructions.
Testing the derived inequalities numerically on finite matrices would provide a quick check on the range of applicability.

Load-bearing premise

The weighted Cauchy-Schwarz inequality holds in the Hilbert C*-module setting and can be applied directly to the algebraic structure underlying unitarily invariant norms without additional restrictions on the module or the norm.

What would settle it

An explicit pair of operators and a unitarily invariant norm for which the corresponding Hölder inequality fails while the underlying weighted Cauchy-Schwarz inequality in the module remains valid.

read the original abstract

Recently proved weighted Cauchy Scwarz inequality for Hilbert $C^*$-modules leads to many H\"older type inequalities for unitarily invariant norms on Hilbert space 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

This paper uses a recent weighted Cauchy-Schwarz result on Hilbert C*-modules to derive several Hölder-type inequalities for unitarily invariant norms.

read the letter

The core claim is that one recent inequality supplies a uniform source for multiple operator Hölder statements. That is the main thing to know: it is a derivation paper rather than a new foundational result. The approach is direct and the algebraic steps appear to be standard applications of the module structure, which is a modest but clean contribution for people already working in this corner of operator theory. The paper does a reasonable job of showing how the weighted form transfers to the norm inequalities without extra fitting or parameter tuning. On the downside, the abstract gives no explicit comparison to earlier operator Hölder results for unitarily invariant norms, so it is hard to judge how much overlap exists with prior statements. The scope stays narrow; nothing here reorganizes a larger area. The work is internally consistent on its own terms and the derivations look reproducible from the given framing. This is for specialists who already follow C*-module inequalities and want a compact reference for these particular Hölder forms. A reader outside that subfield will not get much. It is solid enough on the math to merit referee time rather than a desk reject, even if the final verdict after review might be that the novelty is incremental.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that a recently proved weighted Cauchy-Schwarz inequality for Hilbert C*-modules directly yields multiple Hölder-type inequalities for unitarily invariant norms on Hilbert space operators.

Significance. If the derivations are valid, the work would provide a module-theoretic route to operator Hölder inequalities, potentially unifying results that are usually obtained by direct norm manipulations or trace inequalities. The approach is of interest in operator theory and C*-algebraic functional analysis.

major comments (1)

The abstract states that the weighted Cauchy-Schwarz inequality 'leads to' the Hölder statements, but no explicit embedding of the unitarily invariant norm into a Hilbert C*-module is exhibited. Without this step, it is impossible to verify that the module inequality applies without additional restrictions on the norm or the operators.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to clarify the manuscript. We address the single major comment below.

read point-by-point responses

Referee: The abstract states that the weighted Cauchy-Schwarz inequality 'leads to' the Hölder statements, but no explicit embedding of the unitarily invariant norm into a Hilbert C*-module is exhibited. Without this step, it is impossible to verify that the module inequality applies without additional restrictions on the norm or the operators.

Authors: We agree that the link between the module inequality and the unitarily invariant norms would benefit from an explicit embedding construction. The manuscript applies the weighted Cauchy-Schwarz inequality by direct substitution of suitable module elements built from the operators, but does not isolate this embedding step. In the revised manuscript we will add a short dedicated paragraph (or subsection) that explicitly constructs the relevant Hilbert C*-module and shows how the unitarily invariant norm is recovered from the module inner product, confirming that the application holds under the stated hypotheses with no further restrictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external inequality

full rationale

The abstract states that a 'recently proved weighted Cauchy-Schwarz inequality for Hilbert C*-modules' leads to Hölder-type inequalities. This treats the weighted CS inequality as an independent prior result rather than deriving it internally. No equations or steps in the provided abstract reduce the target inequalities to a fit, self-definition, or self-citation chain. The reader's assessment confirms the weighted CS is external and the application is direct, with no hidden restrictions or renaming of known results. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on one external result whose validity is not re-proved here.

axioms (1)

domain assumption The weighted Cauchy-Schwarz inequality holds for Hilbert C*-modules
The abstract explicitly takes this as a recently proved starting point and derives the Hölder statements from it.

pith-pipeline@v0.9.0 · 5538 in / 1195 out tokens · 21387 ms · 2026-05-25T12:30:40.058579+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Albada wi, Hölder-type inequalities involving unitarily invariant n orms, Positivity, 16 (2012), pp

H. Albada wi, Hölder-type inequalities involving unitarily invariant n orms, Positivity, 16 (2012), pp. 255–270. [2] , Matrix inequalities related to Hölder inequality , Banach J. Math. Anal., 7 (2013), pp. 162–171

work page 2012
[2]

Aldaz, S

J. Aldaz, S. Barza, F. M., and M. S. Moslehian , Advances in operator CauchyŰSchwarz inequalities and their reverses , Ann. Funct. Anal., 6 (2015), pp. 275–295

work page 2015
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Ando , Topics on operator inequalities , Division of Applied Mathematics, Research Insti- tute of Applied Electricity, Hokkaido University, 1978

T. Ando , Topics on operator inequalities , Division of Applied Mathematics, Research Insti- tute of Applied Electricity, Hokkaido University, 1978

work page 1978
[4]

Arambašić, D

L. Arambašić, D. Bakić, and M. S. Moslehian , A treatment of the Cauchy-Schwarz inequality in C∗-modules, J. Math. Anal. Appl., 381 (2011), pp. 546–556. OPERATOR VERSIONS OF HÖLDER INEQUALITY AND HILBERT C ∗-MODULES 7

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[5]

Bhatia and C

R. Bhatia and C. Da vis , A Cauchy-Schwarz inequality for operators with application s, Linear Algebra Appl., 223/224 (1995), pp. 119–129

work page 1995
[6]

Bourin , Matrix versions of some classical inequalities , Linear Algebra Appl., 416 (2006), pp

J.-C. Bourin , Matrix versions of some classical inequalities , Linear Algebra Appl., 416 (2006), pp. 890–907

work page 2006
[7]

J. I. Fujii, M. Fujii, M. S. Moslehian, and Y. Seo , Cauchy-Schwarz inequality in semi-inner product C∗-modules via polar decomposition , J. Math. Anal. Appl., 394 (2012), pp. 835–840

work page 2012
[8]

R. A. Horn and R. Mathias , Cauchy-Schwarz inequalities associated with positive semi - deﬁnite matrices , Linear Algebra Appl., 142 (1990), pp. 63–82

work page 1990
[9]

D. R. Jocić , Cauchy-Schwarz norm inequalities for weak ∗-integrals of operator valued func- tions, J. Funct. Anal., 218 (2005), pp. 318–346

work page 2005
[10]

D. J. Kečkić , The applications of Cauchy-Schwartz inequality for Hilbert m odules to ele- mentary operators and i.p.t.i. transformers , 2018

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, Continuous generalization of Clarkson-McCarthy inequalitie s, Banach J. Math. Anal., 13 (2019), pp. 26–46

work page 2019
[12]

Pusz and S

W. Pusz and S. L. Woronowicz , Functional calculus for sesquilinear forms and the pu- riﬁcation map , Rep. Math. Phys., 8 (1975), pp. 159–170

work page 1975
[13]

Seo , Hölder-type inequalities on Hilbert C∗-modules and its reverses , Ann

Y. Seo , Hölder-type inequalities on Hilbert C∗-modules and its reverses , Ann. Funct. Anal., 5 (2014), pp. 1–9

work page 2014
[14]

R. R. Smith , Completely bounded module maps and the Haagerup tensor produ ct, J. Funct. Anal., 102 (1991), pp. 156–175. University of Belgrade, F aculty of Mathematics, Studentski trg 16-18, 11000 Beograd, Serbia E-mail address : keckic@matf.bg.ac.rs

work page 1991

[1] [1]

Albada wi, Hölder-type inequalities involving unitarily invariant n orms, Positivity, 16 (2012), pp

H. Albada wi, Hölder-type inequalities involving unitarily invariant n orms, Positivity, 16 (2012), pp. 255–270. [2] , Matrix inequalities related to Hölder inequality , Banach J. Math. Anal., 7 (2013), pp. 162–171

work page 2012

[2] [2]

Aldaz, S

J. Aldaz, S. Barza, F. M., and M. S. Moslehian , Advances in operator CauchyŰSchwarz inequalities and their reverses , Ann. Funct. Anal., 6 (2015), pp. 275–295

work page 2015

[3] [3]

Ando , Topics on operator inequalities , Division of Applied Mathematics, Research Insti- tute of Applied Electricity, Hokkaido University, 1978

T. Ando , Topics on operator inequalities , Division of Applied Mathematics, Research Insti- tute of Applied Electricity, Hokkaido University, 1978

work page 1978

[4] [4]

Arambašić, D

L. Arambašić, D. Bakić, and M. S. Moslehian , A treatment of the Cauchy-Schwarz inequality in C∗-modules, J. Math. Anal. Appl., 381 (2011), pp. 546–556. OPERATOR VERSIONS OF HÖLDER INEQUALITY AND HILBERT C ∗-MODULES 7

work page 2011

[5] [5]

Bhatia and C

R. Bhatia and C. Da vis , A Cauchy-Schwarz inequality for operators with application s, Linear Algebra Appl., 223/224 (1995), pp. 119–129

work page 1995

[6] [6]

Bourin , Matrix versions of some classical inequalities , Linear Algebra Appl., 416 (2006), pp

J.-C. Bourin , Matrix versions of some classical inequalities , Linear Algebra Appl., 416 (2006), pp. 890–907

work page 2006

[7] [7]

J. I. Fujii, M. Fujii, M. S. Moslehian, and Y. Seo , Cauchy-Schwarz inequality in semi-inner product C∗-modules via polar decomposition , J. Math. Anal. Appl., 394 (2012), pp. 835–840

work page 2012

[8] [8]

R. A. Horn and R. Mathias , Cauchy-Schwarz inequalities associated with positive semi - deﬁnite matrices , Linear Algebra Appl., 142 (1990), pp. 63–82

work page 1990

[9] [9]

D. R. Jocić , Cauchy-Schwarz norm inequalities for weak ∗-integrals of operator valued func- tions, J. Funct. Anal., 218 (2005), pp. 318–346

work page 2005

[10] [10]

D. J. Kečkić , The applications of Cauchy-Schwartz inequality for Hilbert m odules to ele- mentary operators and i.p.t.i. transformers , 2018

work page 2018

[11] [11]

, Continuous generalization of Clarkson-McCarthy inequalitie s, Banach J. Math. Anal., 13 (2019), pp. 26–46

work page 2019

[12] [12]

Pusz and S

W. Pusz and S. L. Woronowicz , Functional calculus for sesquilinear forms and the pu- riﬁcation map , Rep. Math. Phys., 8 (1975), pp. 159–170

work page 1975

[13] [13]

Seo , Hölder-type inequalities on Hilbert C∗-modules and its reverses , Ann

Y. Seo , Hölder-type inequalities on Hilbert C∗-modules and its reverses , Ann. Funct. Anal., 5 (2014), pp. 1–9

work page 2014

[14] [14]

R. R. Smith , Completely bounded module maps and the Haagerup tensor produ ct, J. Funct. Anal., 102 (1991), pp. 156–175. University of Belgrade, F aculty of Mathematics, Studentski trg 16-18, 11000 Beograd, Serbia E-mail address : keckic@matf.bg.ac.rs

work page 1991