1P.G. Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar-751004, India.
2National Institute of Technology Raipur, India
Receive Date: 30 April 2018,
Revise Date: 16 June 2018,
Accept Date: 19 June 2018
Abstract
Several numerical radius inequalities for operator matrices are proved by generalizing earlier inequalities. In particular, the following inequalities are obtained: if $n$ is even, \\ $$2w(T) \leq max\{\| A_1 \|, \| A_2 \|,\ldots, \| A_n \|\}+\frac{1}{2}\displaystyle\sum_{k=0}^{n-1} \|~ |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} \|$$\\ and if $n$ is odd, \\ $$2w(T) \leq max\{\| A_1 \|,\| A_2 \|,\ldots,\| A_n \|\}+ w\bigg(\widetilde{A}_{(\frac{n+1}{2})t}\bigg)+ \frac{1}{2}\displaystyle\sum_{k=0}^{n-1} \|~ |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} \|$$ for all $t\in [0, 1]$, $ A_i$'s are bounded linear operators on the Hilbert space $\mathcal{H}$, and $ T=\begin{bmatrix} {\bigzero} & & A_1\\ & \hspace{4mm}A_2 & \\ & \hspace{-8mm}\reflectbox{$\ddots$} & \\ A_n & & {\!\!\bigzero} \end{bmatrix}$.
Top