Infinite dimensional Hilbert tensors on spaces of analytic functions

TL;DR

This paper introduces infinite dimensional Hilbert tensors and associated operators on analytic function spaces, establishing their boundedness and norm bounds on Bergman spaces for specific parameter ranges.

Contribution

It defines Hilbert tensor operators on analytic functions and derives their boundedness and explicit norm bounds on Bergman spaces, extending tensor analysis to infinite dimensions.

Findings

Hilbert tensor operators are bounded on Bergman spaces for p > 2(m-1)

Explicit upper bounds for operator norms are established as π and π^{1/(m-1)}

Norms are specifically bounded on A^{4(m-1)} spaces

Abstract

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.