Development of inequality and characterization of equality conditions for the numerical radius
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866917746923536384 |
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| author | Bhunia, Pintu Paul, Kallol |
| author_facet | Bhunia, Pintu Paul, Kallol |
| contents | Let $A$ be a bounded linear operator on a complex Hilbert space and $\Re(A)$ ( $\Im(A)$ ) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of $A$, we prove that \begin{eqnarray*} w(A)&\geq &\frac{1}{2} \left \|A \right\| + \frac{ 1}{2} \mid \|\Re(A)\|-\|\Im(A)\|\mid,\,\,\mbox{and}\\ w^2(A)&\geq& \frac{1}{4} \left \|A^*A+AA^* \right\| + \frac{1}{2}\mid \|\Re(A)\|^2-\|\Im(A)\|^2 \mid, \end{eqnarray*} where $w(A)$ is the numerical radius of the operator $A$. We study the equality conditions for $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ and prove that $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|} $ if and only if the numerical range of $A$ is a circular disk with center at the origin and radius $\frac{1}{2}\sqrt{\|A^*A+AA^*\|} $.
We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_09715 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Development of inequality and characterization of equality conditions for the numerical radius Bhunia, Pintu Paul, Kallol Functional Analysis 47A12, 47A30, 15A60 Let $A$ be a bounded linear operator on a complex Hilbert space and $\Re(A)$ ( $\Im(A)$ ) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of $A$, we prove that \begin{eqnarray*} w(A)&\geq &\frac{1}{2} \left \|A \right\| + \frac{ 1}{2} \mid \|\Re(A)\|-\|\Im(A)\|\mid,\,\,\mbox{and}\\ w^2(A)&\geq& \frac{1}{4} \left \|A^*A+AA^* \right\| + \frac{1}{2}\mid \|\Re(A)\|^2-\|\Im(A)\|^2 \mid, \end{eqnarray*} where $w(A)$ is the numerical radius of the operator $A$. We study the equality conditions for $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ and prove that $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|} $ if and only if the numerical range of $A$ is a circular disk with center at the origin and radius $\frac{1}{2}\sqrt{\|A^*A+AA^*\|} $. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones. |
| title | Development of inequality and characterization of equality conditions for the numerical radius |
| topic | Functional Analysis 47A12, 47A30, 15A60 |
| url | https://arxiv.org/abs/2105.09715 |