Positive definiteness of infinite and finite dimensional generalized Hilbert tensors and generalized Cauchy tensor

An Infinite and finite dimensional generalized Hilbert tensor with a is positive definite if and only if $a>0$. The infinite dimensional generalized Hilbert tensor related operators $F_{\infty }$ and $T_{\infty }$ are bounded, continuous and positively homogeneous. A generalized Cauchy tensor of which generating vectors are $c,d$ is positive definite if and only if every element of vector d is not zero and each element of vector c is positive and mutually distinct. The 4th order n-dimensional generalized Cauchy tensor is matrix positive semi-definite if and only if every element of generating vector c is positive. Finally, the other properties of generalized Cauchy tensor are presented.