$q$-Numerical radius of sectorial matrices and $2 \times 2$ operator matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917242665435136 |
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| author | Rani, Jyoti Patra, Arnab |
| author_facet | Rani, Jyoti Patra, Arnab |
| contents | This article focuses on several significant bounds of $q$-numerical radius $w_q(A)$ for sectorial matrix $A$ which refine and generalize previously established bounds. One of the significant bounds we have derived is as follows:
\[\frac{|q|^2\cos^2α}{2} \|A^*A+AA^*\| \le w_q^2(A)\le \frac{\left(\sqrt{(1-|q|^2)\left(1+2sin^2(α)\right)}+ |q|\right)^2}{2} \|A^*A+AA^*\|,\]
where $ A $ is a sectorial matrix. Also, upper bounds for commutator and anti-commutator matrices and relations between $w_q(A^t)$ and $w_q^t(A)$ for non-integral power $t\in [0,1]$ are also obtained. Moreover, a few significant estimations of $q$-numerical radius of off-diagonal $2\times2$ operator matrices are developed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $q$-Numerical radius of sectorial matrices and $2 \times 2$ operator matrices Rani, Jyoti Patra, Arnab Functional Analysis 15A60 (Primary) 15B48, 47B44 (Secondary) 47A63 This article focuses on several significant bounds of $q$-numerical radius $w_q(A)$ for sectorial matrix $A$ which refine and generalize previously established bounds. One of the significant bounds we have derived is as follows: \[\frac{|q|^2\cos^2α}{2} \|A^*A+AA^*\| \le w_q^2(A)\le \frac{\left(\sqrt{(1-|q|^2)\left(1+2sin^2(α)\right)}+ |q|\right)^2}{2} \|A^*A+AA^*\|,\] where $ A $ is a sectorial matrix. Also, upper bounds for commutator and anti-commutator matrices and relations between $w_q(A^t)$ and $w_q^t(A)$ for non-integral power $t\in [0,1]$ are also obtained. Moreover, a few significant estimations of $q$-numerical radius of off-diagonal $2\times2$ operator matrices are developed. |
| title | $q$-Numerical radius of sectorial matrices and $2 \times 2$ operator matrices |
| topic | Functional Analysis 15A60 (Primary) 15B48, 47B44 (Secondary) 47A63 |
| url | https://arxiv.org/abs/2501.14505 |