Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators
arXiv:2010.15046 · doi:10.1515/ms-2022-0067
Abstract
We give new inequalities for -operator seminorm and -numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space and a non-zero positive bounded linear operator on we show with among other seminorm inequalities, if , i.e., if -adjoint of exist then Further, we prove that if then \begin{eqnarray*} \frac{1}{4}\|T^{\sharp_{A}}T+TT^{\sharp_{A}}\|_A \leq \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg), ~~\textit{and} \end{eqnarray*} \begin{eqnarray*} \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg) +\frac{1}{8}c_A^2\big(T+T^{\sharp_{A}}\big)+\frac{1}{8}c_A^2\big(T-T^{\sharp_{A}}\big) \leq w^2_A(T). \end{eqnarray*} Here and denote -numerical radius, -Crawford number and -operator seminorm, respectively.