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 $a\ge 1$. 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.

15 pages