[Paper Review] Inequalities for the Norm and Numerical Radius of Composite Operators in Hilbert Spaces

Sever S Dragomir|ArXiv.org|Sep 9, 2005

Mathematical Inequalities and Applications4 citations

TL;DR

This paper establishes new inequalities relating the norm and numerical radius of composite operators in Hilbert spaces, particularly for products and sums of bounded linear operators. It derives bounds involving $ \left\|\frac{A^*A + B^*B}{2}\right\| $ and $ w(B^*A) $, with key results showing $ \left\|\frac{A^*A + B^*B}{2}\right\| \leq w(B^*A) + \frac{1}{2}\|A - B\|^2 $, and extends to normal, unitary, and invertible operators under perturbation conditions.

ABSTRACT

Some new inequalities for the norm and the numerical radius of composite operators generated by a pair of operators are given.

Motivation & Objective

• To derive new upper bounds for the norm and numerical radius of composite operators $ B^*A $ in Hilbert spaces.
• To investigate the relationship between $ \left\|\frac{A^*A + B^*B}{2}\right\| $ and $ w(B^*A) $ under operator norm perturbations.
• To extend classical numerical radius inequalities to cases involving invertible, normal, or unitary operators under specific norm constraints.
• To provide quantitative estimates for $ \|A\|\|B\| - w(B^*A) $ and $ \|A\|^2\|B\|^2 - w^2(B^*A) $ under controlled operator differences.

Proposed method

• Establishing operator norm inequalities via the triangle inequality and real part estimation on inner products.
• Using the identity $ \|Ax\|^2 + \|Bx\|^2 \geq 2|\langle B^*A x, x \rangle| $ to relate quadratic forms to numerical radius.
• Applying the condition $ \|A - B\| \leq r $ to derive perturbation bounds involving $ r^2 $, leading to $ \left\|\frac{A^*A + B^*B}{2}\right\| \leq w(B^*A) + \frac{1}{2}r^2 $.
• Utilizing the spectral norm and numerical radius properties of self-adjoint and normal operators to refine bounds.
• Introducing invertibility conditions and operator resolvent estimates to derive bounds involving $ \|B^{-1}\| $, such as $ \|A\|^2\|B\|^2 - w^2(B^*A) \leq 2w(B^*A) \cdot \frac{\|B\|}{\|B^{-1}\|} \left( \|B\|\|B^{-1}\| - \sqrt{1 - r^2\|B^{-1}\|^2} \right) $.
• Applying Cauchy-Schwarz and AM-GM type estimates on inner products to bound differences between norms and numerical radii.

Experimental results

Research questions

• RQ1 How can the norm of $ \frac{A^*A + B^*B}{2} $ be bounded in terms of the numerical radius of $ B^*A $ under a perturbation $ \|A - B\| \leq r $?
• RQ2 What are the tightest possible upper bounds for $ \|A\|\|B\| - w(B^*A) $ and $ \|A\|^2\|B\|^2 - w^2(B^*A) $ when $ A $ and $ B $ are close in operator norm?
• RQ3 How do the bounds change when $ A $ and $ B $ are normal, unitary, or invertible, especially under $ \|A - \lambda B\| \leq r $?
• RQ4 Can the difference between the norm and numerical radius of composite operators be quantified using spectral and resolvent properties?
• RQ5 What are the implications of the bound $ \|A\|^2\|B\|^2 - w^2(B^*A) \leq 2w(B^*A) \cdot \frac{\|B\|}{\|B^{-1}\|} \left( \|B\|\|B^{-1}\| - \sqrt{1 - r^2\|B^{-1}\|^2} \right) $ for invertible operators?

Key findings

• The inequality $ \left\|\frac{A^*A + B^*B}{2}\right\| \leq w(B^*A) + \frac{1}{2}\|A - B\|^2 $ holds for all bounded linear operators $ A, B $ on a Hilbert space.
• The difference $ \left\|\frac{A^*A + B^*B}{2}\right\| - w(B^*A) $ is bounded above by $ \frac{1}{2}\|A - B\|^2 $, with the lower bound being non-negative.
• For a normal operator $ T $ satisfying $ \|T - \lambda T^*\| \leq r $, the inequality $ 0 \leq \frac{1 + |\lambda|^2}{2}\|T\|^2 - |\lambda|w(T^2) \leq \frac{1}{2}r^2 $ holds.
• When $ \|A - \lambda I\| \leq r $, the bound $ 0 \leq \left\|\frac{A^*A + |\lambda|^2 I}{2}\right\| - |\lambda|w(A) \leq \frac{1}{2}r^2 $ is established.
• For invertible $ B $ with $ \|A - B\| \leq r $ and $ \|B^{-1}\| \leq \frac{1}{r} $, the inequality $ \|A\|^2\|B\|^2 - w^2(B^*A) \leq 2w(B^*A) \cdot \frac{\|B\|}{\|B^{-1}\|} \left( \|B\|\|B^{-1}\| - \sqrt{1 - r^2\|B^{-1}\|^2} \right) $ is proven.
• In the special case $ B = \lambda I $, the bound simplifies to $ 0 \leq \|A\|^2 - w^2(A) \leq 2|\lambda|w(A)\left(1 - \sqrt{1 - \frac{r^2}{|\lambda|^2}}\right) $, valid when $ \|A - \lambda I\| \leq r $.