Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices

Kallol paul; Pintu Bhunia; Raj Kumar Nayak

Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every nonzero Hilbert-space operator, the numerical radius is at least half the operator norm plus a Crawford-number correction.

desk verdict The main numerical radius bound (2.3) is correct and genuinely new; the paper deserves a serious referee, though a few secondary inequalities need cleanup. read the letter →

arxiv 1908.04499 v2 pith:VLG2GP5M submitted 2019-08-13 math.FA

Pintu Bhunia , Kallol Paul , Raj kumar Nayak This is my paper · ORCID

classification math.FA MSC 47A1247A6347A30

keywords numericalradiusCrawfordnumberoperatornormmatrixHilbertspacerangeinequalitiesboundedlinear

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves sharper upper and lower estimates for the numerical radius $w(T)$ of a bounded linear operator $T$ on a complex Hilbert space, the largest modulus of an expectation value $\langle Tx,x\rangle$ over unit vectors. Its main lower bound is $w(T)\ge \lVert T\rVert/2 + m(T^2)/(2\lVert T\rVert)$ for every nonzero $T$, where $m(T^2)$ is the Crawford number of $T^2$, the smallest modulus attained by $\langle T^2x,x\rangle$. Because $m(T^2)\ge 0$, this strictly improves the classical inequality $w(T)\ge \lVert T\rVert/2$ whenever $T^2$ has numerical range away from zero. The paper also obtains upper and lower bounds for numerical radii of operator matrices and gives examples where the new bounds beat earlier estimates.

What carries the argument

The central object is the numerical range $W(T)=\{\langle Tx,x\rangle: \lVert x\rVert=1\}$; its extremal radii are the numerical radius $w(T)=\sup\{|\lambda|:\lambda\in W(T)\}$ and the Crawford number $m(T)=\inf\{|\lambda|:\lambda\in W(T)\}$. The argument runs through a three-operator inequality $|\langle A^*TBx,x\rangle|+|\langle B^*TAx,x\rangle|\le 2w(T)\lVert Ax\rVert\lVert Bx\rVert$, proved by expressing the sum of two expectation values as the difference of two numerical-range values and choosing a parameter optimally. Applying this with $A=B=T$ and using the standard identity $w\left(\begin{smallmatrix}0&T\\T&0\end{smallmatrix}\right)=w(T)$ turns the inequality into the lower bound for $T$ alone; the same lemma, specialized to operator matrices, yields the paper's upper and lower bounds for matrix entries.

What would settle it

Search for a nonzero finite matrix $T$ with $w(T)<\lVert T\rVert/2 + m(T^2)/(2\lVert T\rVert)$; the theorem asserts that none exists, so a single such matrix would refute it. A natural check is to take $2\times2$ or $3\times3$ matrices with $m(T^2)>0$ and compute both sides numerically.

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Extended reading notes

Core claim

The paper's central discovery is that the classical lower bound is not tight: the gap between numerical radius and half the operator norm is controlled by the Crawford number of the square. More precisely, for every nonzero $T$, $w(T)\ge \lVert T\rVert/2 + m(T^2)/(2\lVert T\rVert)$, with a second bound $w(T)\ge c^2(T)/(2\lVert T\rVert)+w(T^2)/(2\lVert T\rVert)$ using the minimum norm $c(T)$. The two bounds are combined into a single maximum in Corollary 2.12. For $n\times n$ matrices, the paper further proves that equality $w(T)=\lVert T\rVert/2$ holds exactly when $T$ is unitarily similar to the direct sum of a nilpotent block of norm $\lVert T\rVert$ and a remainder of numerical radius at most $1/2$, so the minimal ratio is attained only by operators whose square has Crawford number zero.

Load-bearing premise

The proof of the main lower bound assumes without proof the standard identity $w\left(\begin{smallmatrix}0&T\\T&0\end{smallmatrix}\right)=w(T)$; if that identity failed, the improvement term $m(T^2)/(2\lVert T\rVert)$ would not follow.

Editorial extensions

If this is right

If correct, the classical estimate $w(T)\ge \lVert T\rVert/2$ is strict for every operator whose square has positive Crawford number.
Any operator attaining the minimal value $w(T)=\lVert T\rVert/2$ must satisfy $m(T^2)=0$, and for matrices this forces the explicit block structure of Theorem 2.10.
The two lower bounds combine to give the stronger estimate $w(T)\ge \frac{1}{2\lVert T\rVert}\max\{\lVert T\rVert^2+m(T^2),\,c^2(T)+w(T^2)\}$.
The new upper bounds for $2\times2$ operator matrices can be strictly better than existing bounds, as the paper's worked examples show.
In applications that estimate norms through the numerical radius, these bounds narrow the interval in which the true operator norm can lie.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not in the paper: a natural analogue with higher powers is $w(T)\ge \lVert T\rVert/2 + m(T^n)/(2\lVert T\rVert^{n-1})$ for $n>2$, worth testing since the square case arises because the off-diagonal $2\times2$ matrix squares to $T^2$.
Not in the paper: the equality condition suggests that minimal-numerical-radius operators are exactly those whose square has zero in the closure of its numerical range, a characterization that could be linked to existing results on numerical-radius attainability.
Not in the paper: for finite matrices the bound supplies a cheap numerical lower test to screen matrices for near-minimal numerical radius in optimization or randomized searches.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline result is the lower bound w(T) ≥ ‖T‖/2 + m(T²)/(2‖T‖) for nonzero T on a Hilbert space. It is correct. I checked the proof and the stress-test note is right: the equality w([[0,T],[T,0]]) = w(T) is standard and true, and the bound can even be derived directly from Lemma 2.1. This is a genuine improvement over the classical 1963 inequality whenever m(T²) > 0, and it gives a clean necessary condition for equality in the old bound. The paper does this honestly, with a self-contained proof using the parallelogram law.

What else is new: the authors prove several upper and lower bounds for operator matrices, compare them with existing bounds, and give explicit numerical examples. They are also transparent that Theorem 2.14 was already proved by Hirzallah et al. The method is not a new technology; it is a careful refinement of known techniques, but the main bound is new and useful.

Where the soft spots are, and they are minor. Corollary 2.5's second displayed inequality does not obviously follow from the stated substitutions into Theorem 2.4. It is likely a typo or a missing intermediate step, but a referee should ask the authors to spell it out. Theorem 2.10's proof is deferred entirely to a citation; acceptable for a known characterization, but thin. Theorem 2.16's proof contains a step where |a² − b²| is bounded using quantities involving m(ReT) and m(ImT); that step is not immediate and should be checked. None of these affect the central inequality. There are also some small typos and awkward phrases, but nothing that undermines the mathematics.

The citation pattern looks normal for the subfield. The authors cite their own related work, but the key new inequality is not secretly recycled. No code or data, which is standard for pure functional analysis.

Who this is for: people working on numerical radius inequalities and operator matrices. It will be a useful reference for the improved lower bound and for comparison inequalities. It is not a blockbuster, but it is a solid, honest piece of work.

My recommendation: yes, send it to peer review. A careful referee can fix the secondary issues and the paper will make a decent contribution to the literature.

Referee Report

0 major / 5 minor

Summary. The paper presents new upper and lower bounds for the numerical radius of bounded Hilbert space operators and of operator matrices. Section 2 generalizes the Bernau–Smithies parallelogram-law inequality to three operators (Lemma 2.2), then derives product bounds (Theorem 2.4, Corollary 2.5), lower bounds for 2×2 operator matrices (Theorem 2.7), and the main result Theorem 2.8: for every nonzero T, w(T) ≥ ‖T‖/2 + m(T²)/(2‖T‖), together with a companion inequality involving w(T²) and c(T). Section 3 gives upper bounds for n×n and 2×2 operator matrices, and Section 4 gives lower bounds, with numerical examples comparing the bounds to earlier ones.

Significance. The central inequality of Theorem 2.8 is a genuine, strict improvement of the classical lower bound w(T) ≥ ‖T‖/2 whenever m(T²) > 0, and it is derived from first principles without free parameters or circular use of the target result. The operator-matrix bounds are concrete and are illustrated by numerical examples. The proof of the main inequality is checkable and self-contained except for one standard unitary-invariance fact; the remaining concerns are local clarity issues rather than correctness problems.

minor comments (5)

[Theorem 2.8] The proof uses the equality w([[0,T],[T,0]]) = w(T) without proof or reference. Since this equality is the bridge between Theorem 2.7 and the main bound (2.3), please add a one-line proof (e.g., via the unitary U = 2^{-1/2}[[I,I],[I,-I]]) or a precise citation.
[Corollary 2.5] The sentence “Taking B = I, T = A and A = B” explains the first displayed inequality but not the second. The second inequality follows from the first inequality of Theorem 2.4 with (A,T,B) = (A*, B*, I), using w((AB)*)=w(AB) and m(A*B*)=m(BA*); please correct the derivation sentence or state the substitution explicitly.
[Theorem 2.10] The proof of Theorem 2.10 is reduced to a citation for the necessity direction and the word “obvious” for sufficiency. Because this is a full characterization, please either quote the exact statement from [8, Th. 1.3-5] or provide a self-contained proof of the necessity direction.
[Theorem 3.7] In the displayed computation of Re²(e^{iθ}T)+Im²(e^{iθ}T), the top-left block of the first RHS matrix is written as Re²(e^{iθ}T)+Im²(e^{iθ}T); it should be Re²(e^{iθ}A)+Im²(e^{iθ}A).
[Theorem 4.1] The proof uses the power inequality w(S²) ≤ w(S)² for the numerical radius, for S = T and S = U*TU, without proof or citation. Please add a reference to this standard fact or include a brief proof.

Circularity Check

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No circularity: the main lower bound (2.3) is derived from the classical Bernau–Smithies inequality and unitary invariance, not from its own conclusion.

full rationale

The central claim of Theorem 2.8, inequality (2.3), is obtained by applying the classical Bernau–Smithies inequality (Lemma 2.1) to the 2x2 operator matrix [[0,T],[T,0]] and then using the unitary-invariance identity w([[0,T],[T,0]]) = w(T). The Crawford-number term m(T^2) enters from the |<T^2x,x>| term in Lemma 2.1; it is not fitted or chosen after computing w(T), and the proof produces a strictly stronger lower bound rather than presupposing it. The unproved equality used in the proof is standard and can be verified by the unitary U = (1/sqrt(2))[[I,I],[I,-I]], so it is not a circular bridge. Lemma 3.1 is cited from the authors' prior work [4], but it is a known Cartesian-decomposition bound used only for the upper-bound results in Section 3; the main lower-bound improvement does not depend on it. The paper explicitly credits Theorem 2.14 to Hirzallah et al. [9], and no parameter is fitted to data anywhere. Thus no derivation step reduces by construction to its own input or to a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities appear. The results rely on standard Hilbert space facts plus one cited bound from the authors' prior paper [4] that is used for upper matrix bounds, not for the headline lower bound.

assumptions (5)

standard math Parallelogram law holds in complex Hilbert spaces and |<Sx,x>| <= w(S)||x||^2 for every operator S.
Used in the core inequality Lemma 2.2 (Eq. 2.2) to bound a difference of quadratic forms by the numerical radius.
standard math The numerical radius of the 2x2 operator matrix [[0,T],[T,0]] equals w(T).
Invoked in Theorem 2.8 to convert the matrix inequality into the lower bound for T itself.
standard math Weak unitary invariance of the numerical radius: w(U*TU)=w(T) for any unitary U.
Used throughout Sections 3 and 4 when permuting rows and columns of operator matrices.
standard math Power inequality w(S^2) <= w(S)^2.
Used in the proof of Theorem 4.1 to pass from w(T^2+(U*TU)^2) to 2w^2(T).
domain assumption Known bound w^2(T) <= ||Re(T)||^2 + ||Im(T)||^2, proved in the authors' earlier paper [4, Remark 2.8].
Used in Lemma 3.1 and Theorem 3.2; cited but not reproved in this paper.

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Cite this review

Pith. "Pith review of Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices." pith.science (2026). https://pith.science/paper/VLG2GP5M

@misc{pith2026190804499,
  author       = {Pith},
  title        = {Pith review of: Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLG2GP5M}},
  note         = {Machine review of arXiv:1908.04499}
}

abstract

We present new upper and lower bounds for the numerical radius of a bounded linear operator defined on a complex Hilbert space, which improve on the existing bounds. Among many other inequalities proved in this article, we show that for a non-zero bounded linear operator $T$ on a Hilbert space $H,$ $w(T)\geq \frac{\|T\|}{2}+\frac{m(T^2)}{2\|T\|}, $ where $w(T)$ is the numerical radius of $T$ and $m(T^2)$ is the Crawford number of $T^2$. This substantially improves on the existing inequality $w(T)\geq \frac{\|T\|}{2} .$ We also obtain some upper and lower bounds for the numerical radius of operator matrices and illustrate with numerical examples that these bounds are better than the existing bounds.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On inequalities for A-numerical radius of operators
math.FA 2019-08 conditional novelty 5.0 of 10

New A-numerical radius bounds are proved for operators, products, and 2x2 operator matrices in semi-Hilbertian spaces, improving on Zamani's 2019 inequalities.

Reference graph

Works this paper leans on

15 extracted references · 8 canonical work pages · cited by 1 Pith paper

[1]

S. Bag, P. Bhunia and K. Paul, Bounds of numerical radius of boun ded linear operator using t-Aluthge transform, arXiv:1904.12096 [math.F A]

arXiv 1904
[2]

Bernau and F

S.J. Bernau and F. Smithies, A note on normal operators, Proc. Cambridge Philos. Soc. 59 (1963) 727-729

1963
[3]

Bhatia, Matrix Analysis, Springer, New York, 1997

R. Bhatia, Matrix Analysis, Springer, New York, 1997

1997
[4]

Bhunia, S

P. Bhunia, S. Bag and K. Paul, Numerical radius inequalities and its a pplications in esti- mation of zeros of polynomials, Linear Algebra Appl. 573 (2019) 166- 177

2019
[5]

Bhunia, S

P. Bhunia, S. Bag and K. Paul, Numerical radius inequalities of oper ator matrices with applications, Linear Multilinear Algebra, (2019) DOI: 10.1080/03081 087.2019.1634673

work page arXiv 2019
[6]

Dragomir, Inequalities for the numerical radius of linear oper ators in Hilbert spaces, Springer, 2013

S.S. Dragomir, Inequalities for the numerical radius of linear oper ators in Hilbert spaces, Springer, 2013

work page 2013
[7]

Guelfen and F

H. Guelfen and F. Kittaneh, On numerical radius inequalities for op erator matrices, Numer. Funct. Anal. Optim. 40 (2019) 1231-1241

work page 2019
[8]

Gustafson and D.K.M

K.E. Gustafson and D.K.M. Rao, Numerical range, Springer, New Y ork, 1997

work page 1997

Show all 15 references

[9]

Hirzallah, F

O. Hirzallah, F. Kittaneh and K. Shebrawi, Numerical radius inequa lities for certain 2 × 2 operator matrices, Integral Equations Operator Theory 71 (20 11) 129-147
[10]

Kittaneh, M.S

F. Kittaneh, M.S. Moslehian and T. Yamazaki, Cartesian decompo sition and numerical radius inequalities, Linear Algebra Appl. 471 (2015) 46-53

2015
[11]

Kittaneh, Numerical radius inequalities for Hilbert spaces ope rators, Studia Math

F. Kittaneh, Numerical radius inequalities for Hilbert spaces ope rators, Studia Math. 168(1) (2005) 73-80. SHARP INEQUALITIES FOR THE NUMERICAL RADIUS 17

2005
[12]

Paul and S

K. Paul and S. Bag, Estimation of bounds for the zeros of a poly nomial using numerical radius, Appl. Math. Comput. 222 (2013) 231-243

2013
[13]

Paul and S

K. Paul and S. Bag, On the numerical radius of a matrix and estim ation of bounds for zeros of a polynomial, Int. J. Math. Math. Sci. 2012 (2012) Art icle Id 129132, DOI:10.1155/1012/129132

2012 doi
[14]

Shebrawi, Numerical radius inequalities for certain 2 ×2 operator matrices II, Linear Algebra Appl

K. Shebrawi, Numerical radius inequalities for certain 2 ×2 operator matrices II, Linear Algebra Appl. 523 (2017) 1-12

2017
[15]

Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math

T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178(1) (2007) 83-89. (Bhunia) Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India E-mail address : pintubhunia5206@gmail.com (Paul) Department ...

2007

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

S. Bag, P. Bhunia and K. Paul, Bounds of numerical radius of boun ded linear operator using t-Aluthge transform, arXiv:1904.12096 [math.F A]

arXiv 1904

[2] [2]

Bernau and F

S.J. Bernau and F. Smithies, A note on normal operators, Proc. Cambridge Philos. Soc. 59 (1963) 727-729

1963

[3] [3]

Bhatia, Matrix Analysis, Springer, New York, 1997

R. Bhatia, Matrix Analysis, Springer, New York, 1997

1997

[4] [4]

Bhunia, S

P. Bhunia, S. Bag and K. Paul, Numerical radius inequalities and its a pplications in esti- mation of zeros of polynomials, Linear Algebra Appl. 573 (2019) 166- 177

2019

[5] [5]

Bhunia, S

P. Bhunia, S. Bag and K. Paul, Numerical radius inequalities of oper ator matrices with applications, Linear Multilinear Algebra, (2019) DOI: 10.1080/03081 087.2019.1634673

work page arXiv 2019

[6] [6]

Dragomir, Inequalities for the numerical radius of linear oper ators in Hilbert spaces, Springer, 2013

S.S. Dragomir, Inequalities for the numerical radius of linear oper ators in Hilbert spaces, Springer, 2013

work page 2013

[7] [7]

Guelfen and F

H. Guelfen and F. Kittaneh, On numerical radius inequalities for op erator matrices, Numer. Funct. Anal. Optim. 40 (2019) 1231-1241

work page 2019

[8] [8]

Gustafson and D.K.M

K.E. Gustafson and D.K.M. Rao, Numerical range, Springer, New Y ork, 1997

work page 1997