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# Upper bounds for Z$_1$-eigenvalues of generalized Hilbert tensors

Submit Time: 2017-12-12
Author: 孟娟 河南师范大学 ; 宋义生 河南师范大学 ;
Institute: 1.河南师范大学;

## Abstracts

 In this paper, we introduce the concept of Z$_1$-eigenvalue to infinite dimensional generalized Hilbert tensors (hypermatrix) $\mathcal{H}_\lambda^{\infty}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$\mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}+\lambda},\ \lambda\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=0,1,2,\cdots,n,\cdots,$$ and proved that its $Z_1$-spectral radius is not larger than $\pi$ for $\lambda>\frac{1}{2}$, and is at most $\frac{\pi}{\sin{\lambda\pi}}$ for $\frac{1}{2}\geq \lambda>0$. Besides, the upper bound of $Z_1$-spectral radius of an $m$th-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_\lambda^n$ is obtained also, and such a bound only depends on $n$ and $\lambda$.
From: 宋义生
DOI：10.12074/201712.02142
Recommended references： 孟娟,宋义生.(2017).Upper bounds for Z$_1$-eigenvalues of generalized Hilbert tensors.[ChinaXiv:201712.02142] (Click&Copy)
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