Refined numerical radius estimates and Euclidean operator radius
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866918369009074176 |
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| author | Bhunia, Pintu Majeed, Rukaya |
| author_facet | Bhunia, Pintu Majeed, Rukaya |
| contents | We obtain new lower and upper bounds for the numerical radius of a bounded linear operator $A$ on a complex Hilbert space, which refine the existing ones.
In particular, if $w(A)$ and $\|A\|$ denote the numerical radius and operator norm of $A$, respectively, then we show that
\begin{eqnarray*}
ν(A) + \frac{1}{4} \left\||A|^2+|A^*|^2\right\|
\leq w^2(A) \leq \frac12 w\left(\frac{|A|+|A^*|}{2}A \right)+ \frac14 \left\| |A|^2+ \left( \frac{|A|+|A^*|}{2}\right)^2 \right\|,
\end{eqnarray*}
where $ν(A)\geq 0$ is a real number involving the operator norm of the Cartesian decomposition of $A$. We also develop several new numerical radius inequalities for the products and sums of operators via Euclidean operator radius of $2$-tuples of operators.
In addition, we deduce equality characterizations for the inequalities.
As an application, we obtain numerical radius inequalities for the commutators of operators, which improves the Fong and Holbrook's inequality $w(AB\pm BA) \leq 2\sqrt{2} w(A) \|B\|$ [Canadian J. Math. 1983]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03962 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Refined numerical radius estimates and Euclidean operator radius Bhunia, Pintu Majeed, Rukaya Functional Analysis 47A12, 47A30, 15A60 We obtain new lower and upper bounds for the numerical radius of a bounded linear operator $A$ on a complex Hilbert space, which refine the existing ones. In particular, if $w(A)$ and $\|A\|$ denote the numerical radius and operator norm of $A$, respectively, then we show that \begin{eqnarray*} ν(A) + \frac{1}{4} \left\||A|^2+|A^*|^2\right\| \leq w^2(A) \leq \frac12 w\left(\frac{|A|+|A^*|}{2}A \right)+ \frac14 \left\| |A|^2+ \left( \frac{|A|+|A^*|}{2}\right)^2 \right\|, \end{eqnarray*} where $ν(A)\geq 0$ is a real number involving the operator norm of the Cartesian decomposition of $A$. We also develop several new numerical radius inequalities for the products and sums of operators via Euclidean operator radius of $2$-tuples of operators. In addition, we deduce equality characterizations for the inequalities. As an application, we obtain numerical radius inequalities for the commutators of operators, which improves the Fong and Holbrook's inequality $w(AB\pm BA) \leq 2\sqrt{2} w(A) \|B\|$ [Canadian J. Math. 1983]. |
| title | Refined numerical radius estimates and Euclidean operator radius |
| topic | Functional Analysis 47A12, 47A30, 15A60 |
| url | https://arxiv.org/abs/2603.03962 |