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arxiv: 1906.08559 · v1 · pith:6P7MVRGQnew · submitted 2019-06-20 · 🧮 math.FA

More accurate numerical radius inequalities

Mohammad Sababheh , Hamid Reza Moradi This is my paper

Pith reviewed 2026-05-25 19:19 UTC · model grok-4.3

classification 🧮 math.FA
keywords numerical radiusHilbert space operatorsoperator inequalitiesabsolute valuenumerical radius bounds
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The pith

For any bounded operator A on a complex Hilbert space, w²(A) is bounded above by the norm of an integral average of |A| and |A*|, itself bounded by half the norm of |A|² + |A*|².

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

The paper establishes refined inequalities for the numerical radius of bounded linear operators on Hilbert spaces by inserting an intermediate integral expression. It proves that the square of the numerical radius is at most the operator norm of the integral from zero to one of the square of a convex combination of |A| and |A*|, and that this quantity is at most half the norm of |A|² plus |A*|². These bounds extend and tighten some earlier results by using a continuous interpolation between the absolute values of an operator and its adjoint. A reader cares because the numerical radius controls many properties of operators yet is typically harder to compute than the norm, so sharper inequalities improve estimates in stability and spectral questions.

Core claim

If A is a bounded linear operator on a complex Hilbert space, then w²(A) ≤ || ∫₀¹ (t |A| + (1-t) |A*| )² dt || ≤ ½ || |A|² + |A*|² ||, where w(A) denotes the numerical radius and ||A|| the operator norm.

What carries the argument

The integral expression ∫₀¹ (t |A| + (1-t) |A*| )² dt, whose operator norm supplies the middle term that refines the upper bound on w²(A).

Load-bearing premise

The numerical radius, the absolute-value operator |A| = (A*A)^{1/2}, and the operator norm satisfy the positivity and comparison properties needed for the integral to be well-defined and for the two inequalities to hold.

What would settle it

An explicit operator A on a Hilbert space for which the operator norm of the integral is strictly less than w²(A) would disprove the left-hand inequality.

read the original abstract

In this article, we present some new general forms of numerical radius inequalities for Hilbert space operators. The significance of these inequalities follow from the way they extend and refine some known results in this field. Among other inequalities, it is shown that if $A$ is a bounded linear operator on a complex Hilbert space, then \[{{w}^{2}}\left( A \right)\le \left\| \int_{0}^{1}{{{\left( t\left| A \right|+\left( 1-t \right)\left| {{A}^{*}} \right| \right)}^{2}}dt} \right\|\le \frac{1}{2}\left\| \;{{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right\|\] where $w\left( A \right)$ and $\left\| A \right\|$ are the numerical radius and the usual operator norm of $A$, respectively.

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 / 1 minor

Summary. The manuscript presents new general forms of numerical radius inequalities for bounded linear operators on complex Hilbert spaces. The central claim is the chain w²(A) ≤ ||∫₀¹ (t |A| + (1-t) |A*| )² dt || ≤ (1/2) || |A|² + |A*|² ||, asserted to extend and refine known results in the area.

Significance. If the derivations hold, the inequalities supply a parameter-free refinement of the standard bound w(A) ≤ ||(|A| + |A*|)/2|| by inserting an explicit integral average whose norm is controlled by the average of the squares. The approach relies only on standard positivity, monotonicity, and quadratic-form estimates in B(H) and introduces no ad-hoc parameters or circular reductions.

minor comments (1)
  1. The abstract states the main inequality without a one-sentence indication of the key steps (pointwise bound on |<Ax,x>| followed by the integral Cauchy-Schwarz estimate); adding such a clause would improve immediate readability while remaining within the abstract length limit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard operator inequalities

full rationale

The claimed inequality is obtained from the standard pointwise bound |<Ax,x>| ≤ (1/2)(<|A|x,x> + <|A*|x,x>), the integral form of Cauchy-Schwarz, and the quadratic-form comparison <(CD+DC)x,x> ≤ <C²x,x> + <D²x,x>, followed by direct expansion of the integral to (1/6)[C² + D² + (C+D)²] and norm comparison. All steps are algebraic identities or well-known facts in B(H) that do not rely on the paper's own results, fitted parameters, or self-citations. The abstract's reference to 'known results' is not load-bearing for the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests entirely on standard axioms and definitions from functional analysis and operator theory with no free parameters, new entities, or ad-hoc assumptions visible in the abstract.

axioms (1)
  • standard math Standard definitions and properties of bounded linear operators on complex Hilbert spaces, including w(A) = sup_{||x||=1} |<Ax, x>|, |A| = (A* A)^{1/2}, and the operator norm.
    Invoked throughout the inequality statement for any such operator A.

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discussion (0)

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Reference graph

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11 extracted references · 11 canonical work pages · 1 internal anchor

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