New numerical radius inequalities for products of Hilbert space operators

Mohsen Shah Hosseini; Baharak Moosavi

doi:10.56754/0719-0646.2803.565

Authors

Mohsen Shah Hosseini Department of Mathematics, ShQ. C., Islamic Azad University, Shar-e Qods, Iran https://orcid.org/0000-0001-9458-7196
Baharak Moosavi Department of Mathematics, ShQ. C., Islamic Azad University, Shar-e Qods, Iran https://orcid.org/0000-0002-4643-4096

MSC:

47A12, 47A30, 47A63

DOI:

https://doi.org/10.56754/0719-0646.2803.565

Keywords:

Bounded linear operator, Hilbert space, norm inequality, numerical radius

Abstract

In this paper, we prove some inequalities of the operator norm and the numerical radius for Hilbert spaces operators. Among other inequalities, it is shown that if \(A, B \in \mathcal{B}(\mathcal{H})\) and \(\mathrm{Re}(A) > 0\), then

\(\displaystyle \omega(BA) \leq \left(\frac{\omega(B)+\|B\|}{2}+D_{B}\right)\omega(A) \leq \frac{7}{2}\omega(A)\omega(B),\)

where \(D_{B}=\inf_{\lambda\in\mathbb{C}}\|B-\lambda I\|\). Moreover, we prove that if \(A, B \in \mathcal{B}(\mathcal{H})\) and \(A > 0\), then

\(\displaystyle \omega(AB+BA) \leq \frac{3\|A\|-\min_{\lambda\in\sigma(A)}\lambda}{2}\,\omega(B) < \frac{3}{2}\|A\|\omega(B).\)

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