Transpose of Nörlund matrices on the domain of summability matrices

Abstract

Abstract Let E=(En,k)n,k≥0 $E=(E_{n,k})_{n,k\geq 0}$ be an invertible summability matrix with bounded absolute row sums and column sums, and let Ep $E_{p}$ denote the domain of E in the sequence space ℓp $\ell _{p}$ (1≤p<∞) $(1\le p<\infty )$. In this paper, we consider the transpose of Nörlund matrix associated with a nonnegative and nonincreasing sequence as an operator mapping ℓp $\ell _{p}$ into the sequence space Ep $E_{p}$ and establish a general upper estimate for its operator norm, which depends on the ℓ1 $\ell _{1}$-norm of the rows and columns of the matrix E. In particular, we apply our result to domains of some summability matrices such as Fibonacci, Karamata, Euler, and Taylor matrices. Our result is an extension of those given by G. Talebi and M.A. Dehghan (Linear Multilinear Algebra 64(2):196–207, 2016). It also provides some analogues of the results by G. Talebi (Indag. Math. 28(3):629–636, 2017).

Keywords

• Summability matrices
• Fibonacci numbers
• Nörlund matrices