Infinite and finite dimensional generalized Hilbert tensors

Abstract

In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a},\ a\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=1,2,\cdots,n, $$ and show that its $H$-spectral radius and its $Z$-spectral radius are smaller than or equal to $M(a)n^{m-1}$ and $M(a)n^{\frac{m}{2}}$, respectively, here $M(a)$ is a constant only dependent on $a$. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for $a\geq1$. For an $m$-order infinite dimensional generalized Hilbert tensor $\mathcal{H}_{\infty}$ with $a>0$, we prove that $\mathcal{H}_{\infty}$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^{1}$ into $l^{p}\ (1<p<\infty)$. The upper bounds of norm of corresponding positively homogeneous operators are obtained.