Approximation of upper bound for matrix operators on the Fibonacci weighted sequence spaces

Abstract

There is a mistake in the statement and proof of Theorem 3.8 of the paper 'Approximation of upper bound for matrix operators on the Fibonacci weighted sequence spaces' that appeared in this journal (64 (2) (2016) 196-207). In this note, the corrected proof of the revised theorem is presented.

Transpose of Nörlund matrices on the domain of summability matrices

“…For this matrix, the sums of all rows are 1, and by Lemma 2.4 of [7] its column sums are bounded. Hence, applying Theorem 1.1 to the Fibonacci sequence space F p , we have the following result, which was previously obtained in Theorem 3.8 of [15].…”

“…We obtain again a general upper estimate for their operator norms, which depend on the 1 -norm of the columns and the rows of the summability matrix E. In particular, we apply our results to domains of some summability matrices such as Fibonacci, Karamata, Euler, and Taylor matrices. Our result is an extension of Theorem 3.8 in [15] and provides some analogue of those given in [13].…”

“…Theorem 1.1 extends [15], Theorem 3.8, from the Fibonacci sequence space to the domains of invertible summability matrices in p . To see this, consider the Fibonacci sequence space F p defined in [7]:…”

Norms and lower bounds of some matrix operators on Fibonacci weighted difference sequence space

“…Also, Kara and Başarır have defined some new spaces as the matrix domain of F in some classical sequence spaces. Recently, by using this matrix, Talebi and Dehghan have introduced the space of Fibonacci weighted sequence space F w , p consisting of all sequences whose F ‐transforms are in , where 1 ≤ p < ∞ and w = ( w k ) is a decreasing nonnegative sequence of real numbers. Further, they have tried to compute the best upper bound for some known matrix operators T from ℓ p ( w ) into F w , p .…”

Approximation of upper bound for matrix operators on the Fibonacci weighted sequence spaces

“…There is a mistake in the statement and proof of Theorem 3.8 of [1], in which an incorrect upper estimate is obtained for the norm of the transpose of Nörlund matrix as operator from the sequence space p into F p , where F p is the domain of the Fibonacci matrix in p . Before correcting the theorem, we announce that the constant…”