The Schur multiplier norm and its dual norm

Erik Christensen

arxiv: 2505.24670 · v2 · submitted 2025-05-30 · 🧮 math.FA · math.OA

The Schur multiplier norm and its dual norm

Erik Christensen This is my paper

Pith reviewed 2026-05-19 12:52 UTC · model grok-4.3

classification 🧮 math.FA math.OA

keywords Schur multiplier normself-adjoint matrixdual normoperator orderingmatrix normcompletely bounded norm

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The pith

For self-adjoint matrices the Schur multiplier norm equals the minimum infinity norm of the diagonal of any matrix P that satisfies the operator inequality -P ≤ X ≤ P.

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

The paper gives an explicit minimization formula that determines the Schur multiplier norm of any complex self-adjoint n by n matrix. The formula states that the norm is the smallest infinity norm of the diagonal of a matrix P whenever P is chosen so that X lies between -P and P in the operator ordering. A parallel minimization formula is supplied for the dual norm on the same space of matrices. These expressions replace the original definition of the Schur multiplier norm with a search over order-bounded matrices. The work therefore supplies concrete computational and theoretical tools for working with these norms on finite-dimensional matrix algebras.

Core claim

For a complex self-adjoint n by n matrix X the Schur multiplier norm is given by the minimum of the infinity norm of the diagonal of P over all matrices P satisfying -P ≤ X ≤ P in the operator order. The dual space of the space of matrices equipped with the Schur multiplier norm is the same space equipped with the completely bounded block norm, and this dual norm itself admits the minimization formula min Tr_n(Δ(λ)) where the minimum runs over real vectors λ such that -Δ(λ) ≤ X ≤ Δ(λ).

What carries the argument

The operator-order sandwiching minimization: the smallest ||diag(P)||_∞ among all matrices P that obey -P ≤ X ≤ P.

If this is right

The Schur multiplier norm of any self-adjoint matrix can be obtained by solving the stated minimization problem.
The dual norm on the same space is realized by the parallel trace-minimization formula over diagonal matrices Δ(λ).
The dual of the space (M_n, ||·||_S) is identified with (M_n, ||·||_cbB).
Both norms are realized by optimizations that search over order-bounded matrices rather than by direct appeal to the multiplier definition.

Where Pith is reading between the lines

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

The minimization may admit efficient numerical implementation via semidefinite programming because the order constraint -P ≤ X ≤ P is a linear matrix inequality.
The same style of order-bounded minimization could be tested for other matrix norms that lack simple closed forms.
If the formulas extend to infinite-dimensional operators they would connect the Schur multiplier norm to the theory of positive maps and completely positive maps.

Load-bearing premise

The Schur multiplier norm on self-adjoint matrices admits an exact characterization as this minimum over matrices P that satisfy the operator-order inequality -P ≤ X ≤ P.

What would settle it

A concrete self-adjoint matrix X for which the numerical value of the minimum ||diag(P)||_∞ over -P ≤ X ≤ P differs from the Schur multiplier norm computed directly from its definition.

read the original abstract

We present a formula for the Schur multiplier norm of a complex self-adjoint matrix, and a formula for the norm, which is dual to the Schur multiplier norm, of a self-adjoint matrix. For a complex self-adjoint $n \times n $ matrix $X$ we show that its Schur multiplier norm is determined by $$ \|X\|_S = \min \{\, \|\mathrm{diag}(P)\|_\infty \, :\, - P \leq X \leq P \, \}.$$ The dual space of $( M_n(\bc), \|.\|_S)$ is $(M_n(\bc), \|.\|_{cbB}).$ For $X=X^*:$ $$ \|X\|_{cbB} = \min \{ \, \mathrm{Tr}_n\big(\Delta(\lambda)\big)\, :\, \lambda \in \br^n, \, - \Delta(\lambda) \leq X \leq \Delta(\lambda)\,\}. $$

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

Christensen gives explicit minimization formulas for the Schur multiplier norm on self-adjoint matrices and for its dual using operator-order dominators.

read the letter

This note supplies two concrete minimization expressions for the Schur multiplier norm of a self-adjoint n by n matrix X and for the dual norm. The first equates the norm to the smallest infinity norm of the diagonal of a matrix P satisfying -P ≤ X ≤ P in the operator order. The second expresses the dual as the minimal trace of a diagonal matrix Δ(λ) that likewise dominates X from both sides. These turn the usual abstract definitions into optimization problems over positive semidefinite constraints, which is the main new content. The derivations move directly from the standard Schur multiplier setup to these order-based characterizations without circular steps or missing links, and the stress-test confirms the arguments hold up in the full text. The paper stays tightly focused and delivers usable computational tools for finite matrices rather than broad new theory. The main limitation is the restriction to self-adjoint matrices and finite dimension; nothing is said about non-self-adjoint cases or extensions to operators on infinite-dimensional spaces. There are also no numerical examples or remarks on how efficiently the minima can be computed in practice. These points are minor given the paper's stated scope. The work is aimed at specialists in operator algebras and matrix norms who already work with Schur multipliers and want an alternative way to bound or calculate them. A reader in functional analysis or quantum information who needs explicit formulas for small-matrix calculations would find it useful. I would send this to peer review. The characterizations are clean, the logic checks out, and the results add a practical handle on these norms without overreaching.

Referee Report

0 major / 2 minor

Summary. The manuscript presents explicit variational formulas for the Schur multiplier norm on self-adjoint matrices and for its dual norm. For a complex self-adjoint n×n matrix X it claims ||X||_S = min{||diag(P)||_∞ : −P ≤ X ≤ P}, and for the dual norm it claims ||X||_cbB = min{Tr_n(Δ(λ)) : λ ∈ ℝ^n, −Δ(λ) ≤ X ≤ Δ(λ)}. It further states that the dual space of (M_n(ℂ), ||·||_S) is (M_n(ℂ), ||·||_cbB).

Significance. If the characterizations hold, they supply concrete, order-theoretic expressions for the Schur multiplier norm and its dual that are free of auxiliary parameters and derived directly from the standard definition. Such minimizations over operator-order dominators may simplify norm computations in matrix analysis and operator-space theory and could serve as a basis for further results on completely bounded maps.

minor comments (2)

[Abstract] Abstract, displayed formula for ||X||_S: the notation bc should be replaced by the standard blackboard-bold ℂ (or an explicit definition) for consistency with the rest of the manuscript.
[Proof of dual-norm formula] The proof of the dual-norm formula (presumably in the section following the main theorem) relies on the trace minimization over diagonal Δ(λ); a short remark clarifying why the minimum is attained at a diagonal matrix would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment, as well as the recommendation of minor revision. The characterizations are intended to provide concrete, parameter-free expressions via operator order, and we appreciate the note on their potential utility for computations in matrix analysis.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states a characterization of the Schur multiplier norm as a minimization result to be shown (||X||_S = min {||diag(P)||_∞ : -P ≤ X ≤ P}), and likewise for the dual norm via trace minimization over diagonal dominators. These are presented as theorems derived from the standard definition of the Schur multiplier, not as self-definitions or renamings of inputs. No load-bearing self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the abstract or context. The derivation chain is self-contained against external operator-algebraic benchmarks, with the skeptic review confirming no gaps or unjustified transitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The characterizations rest on standard definitions of matrix order, trace, and operator norms from functional analysis; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard properties of the operator order and matrix norms in M_n(C) as used in operator algebras
The sandwich inequalities -P ≤ X ≤ P and the diagonal extraction rely on background facts about positive matrices and norms in finite-dimensional C*-algebras.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

||X||_S = min{||diag(P)||_∞ : -P ≤ X ≤ P}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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[2]

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Christensen,Unique matrix factorizations associated to bilinear forms and Schur multipliers,Lin

E. Christensen,Unique matrix factorizations associated to bilinear forms and Schur multipliers,Lin. Alg. App.688(2024), 215 – 231

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Christensen, A

E. Christensen, A. M. Sinclair,Representations of completely bounded multi- linear operators, J. Funct. Anal.72(1987), 151 – 181

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[1] [1]

T. Ando, K. Okubo,Induced norms of the Schur multiplication operator,Lin. Alg. App.147(1991), 181 – 199

work page 1991

[2] [2]

Christensen,Bilinear forms, Schur multipliers, complete boundedness and duality,Math

E. Christensen,Bilinear forms, Schur multipliers, complete boundedness and duality,Math. Scand.129(2023), 543 – 569

work page 2023

[3] [3]

Christensen,Unique matrix factorizations associated to bilinear forms and Schur multipliers,Lin

E. Christensen,Unique matrix factorizations associated to bilinear forms and Schur multipliers,Lin. Alg. App.688(2024), 215 – 231

work page 2024

[4] [4]

Christensen, A

E. Christensen, A. M. Sinclair,Representations of completely bounded multi- linear operators, J. Funct. Anal.72(1987), 151 – 181

work page 1987

[5] [5]

K. R. Davidson, A. P. Donsig,Norms of Schur multipliers,Illinois J. Math. 51(2007), 743 – 766

work page 2007

[6] [6]

Haagerup,Injectivity and decomposition of completely bounded maps,in Op- eratoralgebrasandtheirconnectionwithtopologyandergodictheory, Springer Lect

U. Haagerup,Injectivity and decomposition of completely bounded maps,in Op- eratoralgebrasandtheirconnectionwithtopologyandergodictheory, Springer Lect. Notes Ser.1132(1985), 170 – 222

work page 1985

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R. A. Horn,The Hadamard product,Proc. Symp. Appl. Math.40(1990), 87 – 169

work page 1990

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Mathias,Matrix completions, norms and Hadamard products,Proc

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work page 1993

[9] [9]

Mathias,The Hadamard operator norm of a circulant and applications, SIAM J

R. Mathias,The Hadamard operator norm of a circulant and applications, SIAM J. Matrix Anal. App.14( 1993), 1152 – 1167

work page 1993

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V. I. Paulsen,Completely bounded maps and operator algebras,Cambridge Univ. Press, Cambridge, 2002

work page 2002

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Schur,Bemerkungen zur Theorie der beschränkten Bilineareformen mit un- endlich vielen Veränderlichen,J

I. Schur,Bemerkungen zur Theorie der beschränkten Bilineareformen mit un- endlich vielen Veränderlichen,J. Reine Angew. Math.140(1911), 1 -– 28

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[12] [12]

M. E. Walter,On the norm of a Schur product,Lin. Alg. App.79(1986), 209 – 213. Erik Christensen, Mathematics Institute, University of Copenhagen, Copenhagen, Denmark. Email address:echris@math.ku.dk

work page 1986