Infinite and finite dimensional Hilbert tensors

For an m-order n-dimensional Hilbert tensor (hypermatrix) Hn=(Hi1i2⋯im), Hn=(Hi1i2⋯im=1i1+i2+⋯+im-m+1,i1,⋯,im=1,2,⋯,n its spectral radius is not larger than nm-1sinπn, and an upper bound of its E-spectral radius is nm2sinπn. Moreover, its spectral radius is strictly increasing and its E-spectral radius is nondecreasing with respect to the dimension n. When the order is even, both infinite and finite dimensional Hilbert tensors are positive definite. We also show that the m-order infinite dimensional Hilbert tensor (hypermatrix)H∞=( Hn=(Hi1i2⋯im) defines a bounded and positively (m-1)-homogeneous operator froml1intolp(1<p<∞), and the norm of corresponding positively homogeneous operator is smaller than or equal to π√6.