Infinite and finite dimensional generalized Hilbert tensors

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, $$andshowthatits$H$-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 $a\ge 1$. For an $m$-order infinite dimensional generalized Hilbert tensor ${\mathcal{H}}_{\mathrm{\infty }}$ with $a>0$, we prove that ${\mathcal{H}}_{\mathrm{\infty }}$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^1$ into $l^p(1<p<\mathrm{\infty })$. The upper bounds of norm of corresponding positively homogeneous operators are obtained.