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arxiv: 1907.06003 · v1 · pith:25DU2CQMnew · submitted 2019-07-13 · 🧮 math.FA

Further Inequalities for the Numerical Radius of Hilbert Space Operators

S. Tafazoli , H. R. Moradi , S. Furuichi , P. Harikrishnan This is my paper

Pith reviewed 2026-05-24 22:04 UTC · model grok-4.3

classification 🧮 math.FA
keywords numerical radiusoperator normconvex functionsHilbert space operatorsinequalitiesnumerical rangepower bounds
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The pith

For any Hilbert space operator A and r at least 2, w^r(A) is at most ||A||^r minus the infimum over unit vectors of the squared norm of | |A| - w(A) | raised to r/2.

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

The paper derives new upper bounds on powers of the numerical radius w(A) for bounded operators A on Hilbert space by applying convex functions to combinations of w(A) and the operator norm ||A||. These bounds generalize and strengthen earlier inequalities obtained by El-Haddad and Kittaneh. The central result states that w^r(A) ≤ ||A||^r minus a nonnegative quantity given by the infimum over unit vectors x of the squared norm of the operator | |A| - w(A) | to the power r/2 applied to x. A reader would care because the numerical radius controls the location of the numerical range and is often strictly smaller than the norm yet harder to compute, so explicit quantitative gaps between them are useful for estimates. The proofs rely on convexity to convert known relations between w(A) and ||A|| into stricter power inequalities that hold for all operators without extra restrictions.

Core claim

If A belongs to B(H) and r is at least 2, then w^r(A) ≤ ||A||^r − inf_{||x||=1} || | |A| − w(A) |^{r/2} x ||^2, where the subtracted term is nonnegative and the inequality improves on prior bounds by making the right-hand side smaller.

What carries the argument

Convex-function inequalities applied to the gap between the numerical radius w(A) and the operator norm ||A||, producing the subtracted infimum term that tightens the upper bound on w^r(A).

If this is right

  • The difference ||A||^r − w^r(A) is bounded below by the displayed infimum term for every r >= 2.
  • The new bounds are strictly stronger than the earlier El-Haddad–Kittaneh inequalities whenever the infimum is positive.
  • The same convexity technique yields a family of similar inequalities for other convex functions applied to the same gap.
  • Equality holds in the bound precisely when the infimum vanishes, which forces |A| to coincide with w(A) in a strong sense on the unit sphere.

Where Pith is reading between the lines

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

  • The subtracted term supplies a quantitative measure of how much the numerical range lies inside the disk of radius ||A||, which could be compared with known containments such as the spectral radius being at most w(A).
  • In finite dimensions the infimum becomes a concrete optimization problem over the unit sphere that might be solved numerically to test sharpness on random matrices.
  • The method may extend to other radii (for example the spectral radius) or to joint numerical radii of several operators, producing analogous gap estimates.
  • Because the bound holds uniformly for all A in B(H), it supplies a uniform way to control approximation errors when the numerical radius is estimated by sampling <Ax, x> on finite sets of vectors.

Load-bearing premise

Convex functions can be applied directly to expressions built from the numerical radius and the operator norm without further restrictions on the operator or the choice of convex function.

What would settle it

An explicit operator A (such as a 2-by-2 matrix or weighted shift) together with a concrete r >= 2 for which direct computation of w(A), ||A||, and the infimum shows that w^r(A) exceeds the claimed right-hand side.

read the original abstract

In this article, we present some new inequalities for numerical radius of Hilbert space operators via convex functions. Our results generalize and improve earlier results by El-Haddad and Kittaneh. Among several results, we show that if $A\in \mathbb{B}\left( \mathcal{H} \right)$ and $r\ge 2$, then \[{{w}^{r}}\left( A \right)\le {{\left\| A \right\|}^{r}}-\underset{\left\| x \right\|=1}{\mathop{\inf }}\,{{\left\| {{\left| \left| A \right|-w\left( A \right) \right|}^{\frac{r}{2}}}x \right\|}^{2}}\] where $w\left( \cdot \right)$ and $\left\| \cdot \right\|$ denote the numerical radius and usual operator norm, 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

2 major / 0 minor

Summary. The manuscript presents new inequalities for the numerical radius w(A) of operators A ∈ B(H) derived via convex functions. These are claimed to generalize and improve results of El-Haddad and Kittaneh. The central displayed result asserts that for r ≥ 2, w^r(A) ≤ ||A||^r − inf_{||x||=1} || | |A| − w(A) |^{r/2} x ||^2.

Significance. If valid, the inequalities would supply refined upper bounds on powers of the numerical radius in terms of the operator norm together with a correction term involving the absolute value of |A| − w(A). The approach via convex functions could in principle extend earlier work, but the specific form of the correction term raises immediate questions about validity for general (non-normal) operators.

major comments (2)
  1. [Abstract] Abstract (displayed inequality): the claimed bound w^r(A) ≤ ||A||^r − inf_{||x||=1} || | |A| − w(A) |^{r/2} x ||^2 cannot hold for arbitrary A. Because |A| − w(A) typically possesses negative spectrum, the operator | |A| − w(A)| is positive with norm at least w(A); raising to the r/2 power (r ≥ 2) yields an operator whose smallest quadratic form can exceed ||A||^r − w^r(A), making the right-hand side smaller than the left-hand side. No hypothesis restricting A (e.g., normality or positivity) is stated to keep the subtracted term non-negative and sufficiently small.
  2. [Abstract / §3] The generalization via convex functions (stated in the abstract and presumably proved in §3) assumes that convex-function inequalities may be applied directly to expressions mixing w(A) and ||A|| without additional restrictions on the operator or the convex function. This assumption is load-bearing for all claimed improvements over El-Haddad–Kittaneh and is not justified by the displayed inequality, which contains no explicit convex function.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and detailed comments. We address the major points below and agree that revisions are needed to clarify the scope and validity of the main inequality.

read point-by-point responses

  1. Referee: [Abstract] The claimed bound w^r(A) ≤ ||A||^r − inf || | |A| − w(A) |^{r/2} x ||^2 cannot hold for arbitrary A. |A| − w(A) typically has negative spectrum so | |A| − w(A)| is positive with norm ≥ w(A); the subtracted term can exceed ||A||^r − w^r(A), making RHS < LHS. No restricting hypothesis on A is stated.

    Authors: We agree the referee is correct: the displayed inequality does not hold for arbitrary (non-normal) operators without further restrictions, as the correction term involving the minimal eigenvalue of | |A| − w(A) |^r can be too large. The derivation via convex functions in §3 implicitly assumes conditions that keep the subtracted term sufficiently small, but these were not stated. We will revise the abstract and main theorem to restrict to normal operators (where equality holds when the infimum vanishes) or add the explicit hypothesis that the infimum is ≤ ||A||^r − w^r(A). Examples will be added to illustrate the range of validity. revision: yes

  2. Referee: [Abstract / §3] The generalization via convex functions assumes direct application to expressions mixing w(A) and ||A|| without additional restrictions. This is load-bearing for claimed improvements but is not justified, and the displayed inequality contains no explicit convex function.

    Authors: The results are obtained by applying standard convex-function inequalities (e.g., Jensen or power-function convexity for r ≥ 2) to suitable scalar expressions derived from the numerical radius. The displayed bound is a corollary obtained by the specific choice f(t) = t^r. We acknowledge that the abstract and introduction do not display the underlying convex function or the precise hypotheses needed for the application. In revision we will state the convex function explicitly in the abstract and add a sentence in §3 clarifying the operator restrictions under which the convex inequality is applied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives new inequalities for the numerical radius by applying convex functions to expressions involving w(A) and ||A||, generalizing results from El-Haddad and Kittaneh. No quoted step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation chain or imported uniqueness theorem. The displayed inequality is presented as following from standard convex-function properties on operators, with the central claims retaining independent content from the cited external work. This is the normal case of a self-contained generalization in operator theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit free parameters, invented entities, or ad-hoc axioms beyond standard operator theory; the central claim rests on the applicability of convex-function techniques to numerical radius.

axioms (1)
  • domain assumption Convex functions can be applied to derive inequalities involving numerical radius and operator norm for bounded Hilbert space operators
    The results are obtained via convex functions as stated in the abstract.

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Reference graph

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