Numerical radius inequalities for operator matrices

Document Type: Special issue: Trends in Operators on Banach Spaces

Authors

1 P.G. Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar-751004, India.

2 National Institute of Technology Raipur, India

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}$.

Keywords

Main Subjects