Norm Inequalities for Hilbert space operators with Applications
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914738547458048 |
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| author | Bhunia, Pintu |
| author_facet | Bhunia, Pintu |
| contents | Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator $A,$ it is shown that \begin{eqnarray*}
\|A\|_{p} &\leq &\left(\textit{rank} \, A\right)^{1/{2p}} \|A\|_{2p} \,\, \leq \,\, \left(\textit{rank} \, A\right)^{{(2p-1)}/{2p^2}} \|A\|_{2p^2}, \quad \textit{for all $p\geq 1 $} \end{eqnarray*}
where $\|\cdot\|_p$ is the Schatten $p$-norm. If $\{ λ_n(A) \}$ is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator $A$, then we show that
\begin{eqnarray*}
\sum_{n} \left|λ_n(A)\right|^{p} &\leq& \frac12 \| A\|_{ p}^{ p} + \frac12 \| A^2\|_{p/2}^{p/2}, \quad \textit{for all $p\geq 2$} \end{eqnarray*}
which improves the classical Weyl's inequality $\sum_{n} \left|λ_n(A)\right|^{p} \leq \| A\|_{ p}^{ p}$ [Proc. Nat. Acad. Sci. USA 1949]. For an $n\times n$ matrix $A$, we show that the function $p\to n^{-{1}/{p}}\|A\|_p$ is monotone increasing on $p\geq 1,$ complementing the well known decreasing nature of $p\to \|A\|_p.$
\indent As an application of these inequalities, we provide an upper bound for the sum of the absolute values of the zeros of a complex polynomial. As another application we provide a refined upper bound for the energy of a graph $G$, namely,
$\mathcal{E}(G) \leq \sqrt{2m\left(\textit{rank Adj(G)} \right)},$ where $m$ is the number of edges, improving on a bound by McClelland in $1971$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Norm Inequalities for Hilbert space operators with Applications Bhunia, Pintu Functional Analysis 15A60, 47A30, 47A12, 26C10, 05C50 Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank operator $A,$ it is shown that \begin{eqnarray*} \|A\|_{p} &\leq &\left(\textit{rank} \, A\right)^{1/{2p}} \|A\|_{2p} \,\, \leq \,\, \left(\textit{rank} \, A\right)^{{(2p-1)}/{2p^2}} \|A\|_{2p^2}, \quad \textit{for all $p\geq 1 $} \end{eqnarray*} where $\|\cdot\|_p$ is the Schatten $p$-norm. If $\{ λ_n(A) \}$ is a listing of all non-zero eigenvalues (with multiplicity) of a compact operator $A$, then we show that \begin{eqnarray*} \sum_{n} \left|λ_n(A)\right|^{p} &\leq& \frac12 \| A\|_{ p}^{ p} + \frac12 \| A^2\|_{p/2}^{p/2}, \quad \textit{for all $p\geq 2$} \end{eqnarray*} which improves the classical Weyl's inequality $\sum_{n} \left|λ_n(A)\right|^{p} \leq \| A\|_{ p}^{ p}$ [Proc. Nat. Acad. Sci. USA 1949]. For an $n\times n$ matrix $A$, we show that the function $p\to n^{-{1}/{p}}\|A\|_p$ is monotone increasing on $p\geq 1,$ complementing the well known decreasing nature of $p\to \|A\|_p.$ \indent As an application of these inequalities, we provide an upper bound for the sum of the absolute values of the zeros of a complex polynomial. As another application we provide a refined upper bound for the energy of a graph $G$, namely, $\mathcal{E}(G) \leq \sqrt{2m\left(\textit{rank Adj(G)} \right)},$ where $m$ is the number of edges, improving on a bound by McClelland in $1971$. |
| title | Norm Inequalities for Hilbert space operators with Applications |
| topic | Functional Analysis 15A60, 47A30, 47A12, 26C10, 05C50 |
| url | https://arxiv.org/abs/2404.01982 |