New lower bound for numerical radius for off-diagonal $2\times 2$ matrices

Subject Areas : Operator theory

B. Moosavi ¹ ( Department of Mathematics, Safadasht Branch, Islamic Azad University, Tehran, Iran )
M. Shah Hosseini ² ( Department of Mathematics, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran )

Received: 2023-11-30 Accepted : 2024-01-30 Published : 2024-03-07

Keywords: Hilbert space, Norm inequality, numerical radius, Bounded linear operator,

Abstract :

New norm and numerical radius inequalities for operators on Hilbert space are given. Among other inequalities, we prove that if $ A, B \in B(H) $, then \[\Vert A \Vert - \frac{3 \Vert A-B^* \Vert }{2} \leq \omega\left(\left[\begin{array}{cc} 0 & A \\ B & 0 \end{array}\right]\right).\] Moreover, $\omega(AB) \leq \frac{3}{2} \Vert Im(A) \Vert \Vert B \Vert + D_{B}\; \omega(A) $. In particular, if $ A $ is self-adjointable, then $\omega(AB) \leq D_{B} \Vert A \Vert$, where $D_{B}=\underset{\lambda \in \mathbb{C}}{\mathop{\inf}}\,\left\| B-\lambda I \right\|$.

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