  In this paper, the $m-$order infinite dimensional Hilbert tensor
(hypermatrix) is intrduced to define an $(m-1)$-homogeneous operator on the
spaces of analytic functions, which is called Hilbert tensor operator. The
boundedness of Hilbert tensor operator is presented on Bergman spaces $A^p$
($p>2(m-1)$). On the base of the boundedness, two positively homogeneous
operators are introduced to the spaces of analytic functions, and hence the
upper bounds of norm of such two operators are found on Bergman spaces $A^p$
($p>2(m-1)$). In particular, the norms of such two operators on Bergman spaces
$A^{4(m-1)}$ are smaller than or equal to $\pi$ and $\pi^\frac1{m-1}$,
respectively.
