Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Infinite-dimensional Hilbert tensors on spaces of analytic functions

Pages: 1897 – 1911

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n7.a5

Authors

Yisheng Song (School of Mathematics and Information Science and Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang, Henan, China)

Liqun Qi (Department of AppliedMathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong)

Abstract

In this paper, the $m$th 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 $A^p (p \gt 2(m-1))$. 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 $A^p (p \gt 2(m-1))$. In particular, the norms of such two operators on Bergman spaces $A^{4(m-1)}$ are smaller than or equal to $\pi$ and $\pi^{\frac{1}{m-1}}$, respectively.

Keywords

Hilbert tensor, analytic function, upper bound, Bergman space, gamma function

2010 Mathematics Subject Classification

30C10, 30H05, 30H10, 30H20, 34B10, 47A52, 47H07, 47H09, 47J10

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This work was supported by the National Natural Science Foundation of P.R. China (Grant No. 11571095, 11601134) and the Hong Kong Research Grant Council (Grant No. PolyU 501212, 501913, 15302114 and 15300715).

Paper received on 23 December 2016.

Paper accepted on 28 June 2017.