Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices
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| Format: | Preprint |
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2019
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| _version_ | 1866911982952644608 |
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| author | Bhunia, Pintu Paul, Kallol Nayak, Raj kumar |
| author_facet | Bhunia, Pintu Paul, Kallol Nayak, Raj kumar |
| contents | We present new upper and lower bounds for the numerical radius of a bounded linear operator defined on a complex Hilbert space, which improve on the existing bounds. Among many other inequalities proved in this article, we show that for a non-zero bounded linear operator $T$ on a Hilbert space $H,$ $w(T)\geq \frac{\|T\|}{2}+\frac{m(T^2)}{2\|T\|}, $ where $w(T)$ is the numerical radius of $T$ and $m(T^2)$ is the Crawford number of $T^2$. This substantially improves on the existing inequality $w(T)\geq \frac{\|T\|}{2} .$ We also obtain some upper and lower bounds for the numerical radius of operator matrices and illustrate with numerical examples that these bounds are better than the existing bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1908_04499 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices Bhunia, Pintu Paul, Kallol Nayak, Raj kumar Functional Analysis Primary 47A12, Secondary 47A63, 47A30 We present new upper and lower bounds for the numerical radius of a bounded linear operator defined on a complex Hilbert space, which improve on the existing bounds. Among many other inequalities proved in this article, we show that for a non-zero bounded linear operator $T$ on a Hilbert space $H,$ $w(T)\geq \frac{\|T\|}{2}+\frac{m(T^2)}{2\|T\|}, $ where $w(T)$ is the numerical radius of $T$ and $m(T^2)$ is the Crawford number of $T^2$. This substantially improves on the existing inequality $w(T)\geq \frac{\|T\|}{2} .$ We also obtain some upper and lower bounds for the numerical radius of operator matrices and illustrate with numerical examples that these bounds are better than the existing bounds. |
| title | Sharp inequalities for the numerical radius of Hilbert space operators and operator matrices |
| topic | Functional Analysis Primary 47A12, Secondary 47A63, 47A30 |
| url | https://arxiv.org/abs/1908.04499 |