The Hilbert $L$-matrix
| Autor: | František Štampach |
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| Rok vydání: | 2021 |
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| DOI: | 10.48550/arxiv.2107.10694 |
| Popis: | We analyze spectral properties of the Hilbert $L$-matrix $$\left(\frac{1}{\max(m,n)+\nu}\right)_{m,n=0}^{\infty}$$ regarded as an operator $L_{\nu}$ acting on $\ell^{2}(\mathbb{N}_{0})$, for $\nu\in\mathbb{R}$, $\nu\neq0,-1,-2,\dots$. The approach is based on a spectral analysis of the inverse of $L_{\nu}$, which is an unbounded Jacobi operator whose spectral properties are deducible in terms of the unit argument ${}_{3}F_{2}$-hypergeometric functions. In particular, we give answers to two open problems concerning the operator norm of $L_{\nu}$ published by L. Bouthat and J. Mashreghi in [Oper. Matrices 15, No. 1 (2021), 47--58]. In addition, several general aspects concerning the definition of an $L$-operator, its positivity, and Fredholm determinants are also discussed. Comment: 31 pages, 6 figures, accepted for publication in the Journal of Functional Analysis |
| Databáze: | OpenAIRE |
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