New lower bound for numerical radius for off-diagonal $2\times 2$ matrices
الموضوعات :B. Moosavi 1 , M. Shah Hosseini 2
1 - Department of Mathematics, Safadasht Branch, Islamic Azad University, Tehran, Iran
2 - Department of Mathematics, Shahr-e-Qods Branch, Islamic Azad University, Tehran, Iran
الکلمات المفتاحية: Hilbert space, Norm inequality, numerical radius, Bounded linear operator,
ملخص المقالة :
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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