Improved inequalities for the numerical radius via Cartesian decomposition
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866929456304619520 |
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| author | Bhunia, Pintu Jana, Suvendu Moslehian, Mohammad Sal Paul, Kallol |
| author_facet | Bhunia, Pintu Jana, Suvendu Moslehian, Mohammad Sal Paul, Kallol |
| contents | We develop various lower bounds for the numerical radius $w(A)$ of a bounded linear operator $A$ defined on a complex Hilbert space, which improve the existing inequality $w^2(A)\geq \frac{1}{4}\|A^*A+AA^*\|$. In particular, for $r\geq 1$, we show that \begin{eqnarray*}\frac{1}{4}\|A^*A+AA^*\|
\leq\frac{1}{2} \left( \frac{1}{2}\|\Re(A)+\Im(A)\|^{2r}+\frac{1}{2}\|\Re(A)-\Im(A)\|^{2r}\right)^{\frac{1}{r}}
\leq w^{2}(A),\end{eqnarray*} where $\Re(A)$ and $\Im(A)$ are the real and imaginary parts of $A$, respectively. Furthermore, we obtain upper bounds for $w^2(A)$ refining the well-known upper bound $w^2(A)\leq \frac{1}{2} \left(w(A^2)+\|A\|^2\right)$. Separate complete characterizations for $w(A)=\frac{\|A\|}{2}$ and $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ are also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_02499 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Improved inequalities for the numerical radius via Cartesian decomposition Bhunia, Pintu Jana, Suvendu Moslehian, Mohammad Sal Paul, Kallol Functional Analysis Primary 47A12, Secondary 15A60, 47A30, 47A50 We develop various lower bounds for the numerical radius $w(A)$ of a bounded linear operator $A$ defined on a complex Hilbert space, which improve the existing inequality $w^2(A)\geq \frac{1}{4}\|A^*A+AA^*\|$. In particular, for $r\geq 1$, we show that \begin{eqnarray*}\frac{1}{4}\|A^*A+AA^*\| \leq\frac{1}{2} \left( \frac{1}{2}\|\Re(A)+\Im(A)\|^{2r}+\frac{1}{2}\|\Re(A)-\Im(A)\|^{2r}\right)^{\frac{1}{r}} \leq w^{2}(A),\end{eqnarray*} where $\Re(A)$ and $\Im(A)$ are the real and imaginary parts of $A$, respectively. Furthermore, we obtain upper bounds for $w^2(A)$ refining the well-known upper bound $w^2(A)\leq \frac{1}{2} \left(w(A^2)+\|A\|^2\right)$. Separate complete characterizations for $w(A)=\frac{\|A\|}{2}$ and $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ are also given. |
| title | Improved inequalities for the numerical radius via Cartesian decomposition |
| topic | Functional Analysis Primary 47A12, Secondary 15A60, 47A30, 47A50 |
| url | https://arxiv.org/abs/2110.02499 |