Infinite and finite dimensional generalized Hilbert tensors

TL;DR

This paper introduces generalized Hilbert tensors of arbitrary order and dimension, analyzes their spectral properties, and establishes conditions for positive definiteness and boundedness of associated operators.

Contribution

It defines a new class of generalized Hilbert tensors, derives bounds for their spectral radii, and proves positive definiteness and boundedness of related operators.

Findings

Spectral radii are bounded by functions of tensor size and parameter a.

Generalized Hilbert tensors are positive definite for a ≥ 1.

Infinite-dimensional tensors define bounded operators from l^1 to l^p.

Abstract

In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a},\ a\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=1,2,\cdots,n, $$ and show that its $H$-spectral radius and its $Z$-spectral radius are smaller than or equal to $M(a)n^{m-1}$ and $M(a)n^{\frac{m}{2}}$, respectively, here $M(a)$ is a constant only dependent on $a$. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for $a\geq1$. For an $m$-order infinite dimensional generalized Hilbert tensor $\mathcal{H}_{\infty}$ with $a>0$, we prove that $\mathcal{H}_{\infty}$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^{1}$ into $l^{p}\ (1<p<\infty)$. The upper bounds of norm of corresponding positively homogeneous operators are obtained.