Infinite-dimensional hilbert tensors on spaces of analytic functions

Abstract

In this paper, the mth order infinite dimensional Hilbert tensor (hypermatrix) is introduced to define an (m-1)-homogeneous operator on the spaces of analytic functions, which is called the Hilbert tensor operator. The boundedness of the Hilbert tensor operator is presented on Bergman spaces Ap (p >MergeCell 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 the two operators are found on Bergman spaces Ap (p >MergeCell 2(m-1)). In particular, the norms of such two operators on Bergman spaces A4(m-1) are smaller than or equal to π and π1/m-1, respectively.