## Annals of Functional Analysis

### Upper bounds for numerical radius inequalities involving off-diagonal operator matrices

#### Abstract

In this article, we establish some upper bounds for numerical radius inequalities, including those of $2\times2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=[\begin{smallmatrix}0&X\\Y&0\end{smallmatrix}]$, then $\begin{eqnarray*}\omega^{r}(T)\leq2^{r-2}\Vert f^{2r}(\vert X\vert )+g^{2r}(\vert Y^{*}\vert )\Vert ^{\frac{1}{2}}\Vert f^{2r}(\vert Y\vert )+g^{2r}(\vert X^{*}\vert )\Vert ^{\frac{1}{2}}\end{eqnarray*}$ and $\begin{eqnarray*}\omega^{r}(T)\leq2^{r-2}\Vert f^{2r}(\vert X\vert )+f^{2r}(\vert Y^{*}\vert )\Vert ^{\frac{1}{2}}\Vert g^{2r}(\vert Y\vert )+g^{2r}(\vert X^{*}\vert )\Vert ^{\frac{1}{2}},\end{eqnarray*}$ where $X,Y$ are bounded linear operators on a Hilbert space ${\mathcal{H}}$, $r\geq1$, and $f$, $g$ are nonnegative continuous functions on $[0,\infty)$ satisfying the relation $f(t)g(t)=t$ ($t\in[0,\infty)$). Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators $T_{1},\ldots,T_{n}$.

#### Article information

Source
Ann. Funct. Anal., Volume 9, Number 3 (2018), 297-309.

Dates
Received: 6 June 2017
Accepted: 5 September 2017
First available in Project Euclid: 17 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.afa/1508205625

Digital Object Identifier
doi:10.1215/20088752-2017-0029

Mathematical Reviews number (MathSciNet)
MR3835218

Zentralblatt MATH identifier
06946355

#### Citation

Bakherad, Mojtaba; Shebrawi, Khalid. Upper bounds for numerical radius inequalities involving off-diagonal operator matrices. Ann. Funct. Anal. 9 (2018), no. 3, 297--309. doi:10.1215/20088752-2017-0029. https://projecteuclid.org/euclid.afa/1508205625

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