In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $mathcal{H}_{n}=(mathcal{H}_{i_{1}i_{2}cdots i_{m}})$, $$ mathcal{H}_{i_{1}i_{2}cdots i_{m}}=frac{1}{i_{1}+i_{2}+cdots i_{m}-m+a}, ain mathbb{R}setminusmathbb{Z}^-; i_{1},i_{2},cdots,i_{m}=1,2,cdots,n, $$ 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 $ageq1$. 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.
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