Numerical radius inequalities of sectorial matrices
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866929456342368256 |
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| author | Bhunia, Pintu Paul, Kallol Sen, Anirban |
| author_facet | Bhunia, Pintu Paul, Kallol Sen, Anirban |
| contents | We obtain several upper and lower bounds for the numerical radius of sectorial matrices. We also develop several numerical radius inequalities of the sum, product and commutator of sectorial matrices. The inequalities obtained here are sharper than the existing related inequalities for general matrices. Among many other results we prove that if $A$ is an $n\times n$ complex matrix with the numerical range $W(A)$ satisfying $W(A)\subseteq\{re^{\pm iθ}~:~θ_1\leqθ\leqθ_2\},$ where $r>0$ and $θ_1,θ_2\in \left[0,π/2\right],$ then \begin{eqnarray*}
&&(i)\,\, w(A) \geq \frac{cscγ}{2}\|A\|
+ \frac{cscγ}{2}\left| \|\Im(A)\|-\|\Re(A)\|\right|,\,\,\text{and}
&&(ii)\,\, w^2(A) \geq \frac{csc^2γ}{4}\|AA^*+A^*A\|
+ \frac{csc^2γ}{2}\left| \|\Im(A)\|^2-\|\Re(A)\|^2\right|, \end{eqnarray*} where $γ=\max\{θ_2,π/2-θ_1\}$. We also prove that if $A,B$ are sectorial matrices with sectorial index $γ\in [0,π/2)$ and they are double commuting, then $w(AB)\leq \left(1+\sin^2γ\right)w(A)w(B).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_09816 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Numerical radius inequalities of sectorial matrices Bhunia, Pintu Paul, Kallol Sen, Anirban Functional Analysis 47A12, 47B44, 47A30 We obtain several upper and lower bounds for the numerical radius of sectorial matrices. We also develop several numerical radius inequalities of the sum, product and commutator of sectorial matrices. The inequalities obtained here are sharper than the existing related inequalities for general matrices. Among many other results we prove that if $A$ is an $n\times n$ complex matrix with the numerical range $W(A)$ satisfying $W(A)\subseteq\{re^{\pm iθ}~:~θ_1\leqθ\leqθ_2\},$ where $r>0$ and $θ_1,θ_2\in \left[0,π/2\right],$ then \begin{eqnarray*} &&(i)\,\, w(A) \geq \frac{cscγ}{2}\|A\| + \frac{cscγ}{2}\left| \|\Im(A)\|-\|\Re(A)\|\right|,\,\,\text{and} &&(ii)\,\, w^2(A) \geq \frac{csc^2γ}{4}\|AA^*+A^*A\| + \frac{csc^2γ}{2}\left| \|\Im(A)\|^2-\|\Re(A)\|^2\right|, \end{eqnarray*} where $γ=\max\{θ_2,π/2-θ_1\}$. We also prove that if $A,B$ are sectorial matrices with sectorial index $γ\in [0,π/2)$ and they are double commuting, then $w(AB)\leq \left(1+\sin^2γ\right)w(A)w(B).$ |
| title | Numerical radius inequalities of sectorial matrices |
| topic | Functional Analysis 47A12, 47B44, 47A30 |
| url | https://arxiv.org/abs/2208.09816 |