Infinite and finite dimensional generalized Hilbert tensors

In this paper, we introduce the concept of an m-order n-dimensional generalized Hilbert tensor Hn=(Hi1i2⋯im),Hi1i2⋯im=1i1+i2+⋯im−m+a,a∈R∖Z−;i1,i2,⋯,im=1,2,⋯,n, and show that its H-spectral radius and its Z-spectral radius are smaller than or equal to M(a)nm−1 and M(a)nm2, respectively, here M(a) is a constant depending on a. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for a≥1. For an m-order infinite dimensional generalized Hilbert tensor H∞ with a>0, we prove that H∞ defines a bounded and positively (m−1)-homogeneous operator from l1 into lp(1