[Paper Review] Refined inequalities for the numerical radius of Hilbert space operators

Pintu Bhunia, Suvendu Jana+1 more|arXiv (Cornell University)|Jun 26, 2021

Mathematical Inequalities and Applications16 references5 citations

TL;DR

This paper presents new, tighter upper and lower bounds for the numerical radius of bounded linear operators on complex Hilbert spaces, refining existing inequalities through novel expressions involving the real and imaginary parts of the operator. The key contribution is a sharp lower bound for $ w(A) $ in terms of $ \|A\| $, $ \|\Re(A)\pm\Im(A)\| $, and a refined upper bound for $ w^2(A) $ involving $ \|A^*A + AA^*\| $, which improves upon classical results by Kittaneh and others.

ABSTRACT

We present some new upper and lower bounds for the numerical radius of bounded linear operators on a complex Hilbert space and show that these are stronger than the existing ones. In particular, we prove that if $A$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$ and if $\Re(A)$, $\Im(A)$ are the real part, the imaginary part of $A$, respectively, then $$ w(A)\geq\frac{\|A\|}{2} +\frac{1}{2\sqrt{2}} \Big | \|\Re(A)+\Im(A)\|-\|\Re(A)-\Im(A)\| \Big | $$ and $$ w^2(A)\geq\frac{1}{4}\|A^*A+AA^*\|+\frac{1}{4}\Big| \|\Re(A)+\Im(A)\|^2-\|\Re(A)-\Im(A)\|^2\Big|. $$ Here $w(.)$ and $\|.\|$ denote the numerical radius and the operator norm, respectively. Further, we obtain refinement of inequalities for the numerical radius of the product of two operators. Finally, as an application of the second inequality mentioned above, we obtain an improvement of upper bound for the numerical radius of the commutators of operators.

Motivation & Objective

• To derive tighter upper and lower bounds for the numerical radius of bounded linear operators on complex Hilbert spaces.
• To refine existing inequalities involving the operator norm and numerical radius, particularly those by Kittaneh and Yamazaki.
• To improve bounds for the numerical radius of products and commutators of operators using structural properties of real and imaginary parts.
• To provide a sharper estimate for the numerical radius of generalized commutators $ AXB \pm BYA $, improving on Fong and Holbrook's bound.

Proposed method

• Derives a new lower bound for $ w(A) $ using the triangle inequality and properties of $ \Re(A) \pm \Im(A) $, leveraging the identity $ \langle Ax,x\rangle = \sqrt{\langle\Re(A)x,x\rangle^2 + \langle\Im(A)x,x\rangle^2} $.
• Establishes a refined inequality: $ w(A) \geq \frac{1}{2}\|A\| + \frac{1}{2\sqrt{2}}\left|\|\Re(A)+\Im(A)\| - \|\Re(A)-\Im(A)\|\right| $, which strengthens the classical $ w(A) \geq \frac{1}{2}\|A\| $.
• Introduces a quadratic lower bound: $ w^2(A) \geq \frac{1}{4}\|A^*A + AA^*\| + \frac{1}{4}\left|\|\Re(A)+\Im(A)\|^2 - \|\Re(A)-\Im(A)\|^2\right| $, refining Kittaneh’s inequality.
• Applies these bounds to the product $ B^*A $, deriving a refined inequality for $ w^{2r}(B^*A) $ that improves on Dragomir and Hedarbeygi et al.'s results.
• Uses the refined $ w^2(A) $ bound to derive a sharper upper bound for the numerical radius of commutators $ AXB \pm BYA $, incorporating operator norms and real/imaginary part norms.
• Employs the Cauchy-Schwarz inequality and normalization techniques to extend bounds from contractive to general operators via scaling.

Experimental results

Research questions

• RQ1 Can the classical lower bound $ w(A) \geq \frac{1}{2}\|A\| $ be improved using the real and imaginary parts of $ A $?
• RQ2 How do the norms of $ \Re(A) \pm \Im(A) $ influence the numerical radius of $ A $?
• RQ3 Can the inequality $ w^2(A) \geq \frac{1}{4}\|A^*A + AA^*\| $ be sharpened by incorporating additional terms involving $ \Re(A) \pm \Im(A) $?
• RQ4 Does the new bound for $ w^2(A) $ lead to improved estimates for the numerical radius of operator products and commutators?
• RQ5 Is the bound for $ w(AXB \pm BYA) $ strictly better than existing ones, such as those by Fong and Holbrook or Hirzallah and Kittaneh?

Key findings

• The paper establishes a new lower bound: $ w(A) \geq \frac{1}{2}\|A\| + \frac{1}{2\sqrt{2}}\left|\|\Re(A)+\Im(A)\| - \|\Re(A)-\Im(A)\|\right| $, which strictly improves the classical $ w(A) \geq \frac{1}{2}\|A\| $.
• A refined quadratic bound is proven: $ w^2(A) \geq \frac{1}{4}\|A^*A + AA^*\| + \frac{1}{4}\left|\|\Re(A)+\Im(A)\|^2 - \|\Re(A)-\Im(A)\|^2\right| $, which strengthens Kittaneh’s inequality.
• For the product $ B^*A $, the inequality $ w^{2r}(B^*A) \leq \frac{1}{2}\left(\frac{\||B|^2|A|^2 + |A|^2|B|^2\|}{2}\right)^r + \frac{1}{4}\||B|^{4r} + |A|^{4r}\| $ is derived, refining earlier results by Hedarbeygi et al.
• The bound for the generalized commutator $ w(AXB \pm BYA) $ is improved to $ \leq 2\sqrt{2}\|B\|\max\{\|X\|,\|Y\|\}\sqrt{w^2(A) - \frac{1}{4}\left|\|\Re(A)+\Im(A)\|^2 - \|\Re(A)-\Im(A)\|^2\right|} $, which is tighter than Fong and Holbrook’s bound.
• The new bound for $ w(AXB \pm BYA) $ is not comparable in general with Hirzallah and Kittaneh’s bound, as shown by counterexamples.
• Corollary 3.8 shows that equality in the improved bound occurs only if $ \|\Re(A)+\Im(A)\| = \|\Re(A)-\Im(A)\| $, providing a necessary condition for equality in the refined estimate.