New lower bound for numerical radius for off-diagonal $2\times 2$ matrices
الموضوعات :
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)
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 $D_{B}=\underset{\lambda \in \mathbb{C}}{\mathop{\inf}}\,\left\| B-\lambda I \right\|$.
[1] A. Abu-Omar, F. Kittaneh, Numerical radius inequalities for products of Hilbert space operators, J. Operator Theory. 72 (2) (2014), 521-527.
[2] S. S. Dragomir, Some inequalities for the norm and the numerical radius of linear operators in Hilbert Spaces, Tamkang J. Math. 39 (1) (2008), 1-7.
[3] C. K. Fong, J. A. Holbrook, Unitarily invariant operator norms, Canad. J. Math. 35 (1983), 274-299.
[4] O. Hirzallah, F. Kittaneh, K. Shebrawi, Numerical radius inequalities for certain 2 × 2 operator matrices, Integral Equ. Operator Theory. 71 (2) (2011), 129-147.
[5] J. A. R. Holbrook, Multiplicative properties of the numerical radius in operator theory, J. Reine Angew. Math. 237 (1969), 166-174.
[6] F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168 (1) (2005), 73-80.
[7] M. Shah Hosseini, B. Moosavi, Some numerical radius inequalities for products of Hilbert space operators, Filomat. 33 (7) (2019), 2089-2093.
[8] M. Shah Hosseini, B. Moosavi, H. R. Moradi, An alternative estimate for the numerical radius of Hilbert space operators, Math. Slovaca. 70 (1) (2020), 233-237.
[9] M. Shah Hosseini, M. E. Omidvar, Some inequalities for the numerical radius for Hilbert space operators, Bull. Aust. Math. Soc. 94 (3) (2016), 489-496.
[10] J. G. Stampfli, The norm of a derivation, Pacific J. Math. 33 (1970), 737-747.
[11] T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Stud. Math. 178 (2007), 83-89.
