The norms of certain matrix operators from l_p spaces into l_p(Δⁿ) spaces

ABSTRACT

Let $A={(a_{k,m})}_{k,m\ge 0}$ be a non-negative matrix and ${\mathrm{\Delta }}^n$ be the forward difference operator of order n for $n=1,2,\cdots$. Denote by ${\Vert A\Vert }_{p,{\mathrm{\Delta }}^n}$ the infimum of those U, satisfying the following inequality: $$\begin{array}{c}\hfill {\left(\sum _{k=0}^{\infty }{\left|\sum _{j=0}^n\sum _{m=0}^{\infty }{(-1)}^j\left(\genfrac{}{}{0pt}{}{n}{j}\right)a_{k+j,m}{x}_m\right|}^p\right)}^{\frac{1}{p}}\le U{\left(\sum _{j=0}^{\infty }{|x_j|}^p\right)}^{\frac{1}{p}},\end{array}$$

where $x\in l_p$ and $p\ge 1$. This paper is focused on the problem of finding ${\Vert A\Vert }_{p,{\mathrm{\Delta }}^n}$, where A is a matrix operator from $l_p$ into $l_p({\mathrm{\Delta }}^n)$. In particular, we apply our results to well known matrix operators such as Copson , Hilbert, and Nörlund.

KEYWORDS:

• Matrix operator
• norm
• Copson matrix
• Hilbert matrix
• Nörlund matrix
• difference sequence space

AMS SUBJECT CLASSIFICATIONS:

• 26D15
• 40C05
• 40G05
• 47B37

Notes

No potential conflict of interest was reported by the authors.