Numerical radius inequalities of $2 \times 2$ operator matrices
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866910563865460736 |
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| author | Bhunia, Pintu Paul, Kallol |
| author_facet | Bhunia, Pintu Paul, Kallol |
| contents | Several upper and lower bounds for the numerical radius of $2 \times 2$ operator matrices are developed which refine and generalize the earlier related bounds. In particular, we show that if $B,C$ are bounded linear operators on a complex Hilbert space, then \begin{eqnarray*}
&& \frac{1}{2}\max \left \{ \|B\|, \|C\| \right \}+\frac{1}{4} \left | \|B+C^*\|-\|B-C^*\| \right |
&&\leq w \left(\left[\begin{array}{cc}
0 & B
C& 0
\end{array}\right]\right)\\ &&\leq \frac{1}{2} \max \left\{\|B\|,\|C\|\right \}+\frac{1}{2}\max \left \{r^{\frac{1}{2}}(|B||C^*|),r^{\frac{1}{2}}(|B^*||C|)\right\}, \end{eqnarray*} where $w(.)$, $r(.)$ and $\|.\|$ are the numerical radius, spectral radius and operator norm of a bounded linear operator, respectively. We also obtain equality conditions for the numerical radius of the operator matrix $\left[\begin{array}{cc}
0 & B
C& 0 \end{array}\right]$. As application of results obtained, we show that if $B,C$ are self-adjoint operators then, $\max \Big \{\|B+C\|^2 , \|B-C\|^2 \Big\}\leq \left \|B^2+C^2 \right \|+2w(|B||C|). $ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_09718 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Numerical radius inequalities of $2 \times 2$ operator matrices Bhunia, Pintu Paul, Kallol Functional Analysis 47A12, 47A30 Several upper and lower bounds for the numerical radius of $2 \times 2$ operator matrices are developed which refine and generalize the earlier related bounds. In particular, we show that if $B,C$ are bounded linear operators on a complex Hilbert space, then \begin{eqnarray*} && \frac{1}{2}\max \left \{ \|B\|, \|C\| \right \}+\frac{1}{4} \left | \|B+C^*\|-\|B-C^*\| \right | &&\leq w \left(\left[\begin{array}{cc} 0 & B C& 0 \end{array}\right]\right)\\ &&\leq \frac{1}{2} \max \left\{\|B\|,\|C\|\right \}+\frac{1}{2}\max \left \{r^{\frac{1}{2}}(|B||C^*|),r^{\frac{1}{2}}(|B^*||C|)\right\}, \end{eqnarray*} where $w(.)$, $r(.)$ and $\|.\|$ are the numerical radius, spectral radius and operator norm of a bounded linear operator, respectively. We also obtain equality conditions for the numerical radius of the operator matrix $\left[\begin{array}{cc} 0 & B C& 0 \end{array}\right]$. As application of results obtained, we show that if $B,C$ are self-adjoint operators then, $\max \Big \{\|B+C\|^2 , \|B-C\|^2 \Big\}\leq \left \|B^2+C^2 \right \|+2w(|B||C|). $ |
| title | Numerical radius inequalities of $2 \times 2$ operator matrices |
| topic | Functional Analysis 47A12, 47A30 |
| url | https://arxiv.org/abs/2105.09718 |