Compactness of matrix operators on some sequence spaces derived by Fibonacci numbers

Kara, Emrah Evren; Başarır, Metin; Mursaleen, M.

doi:10.5937/kgjmath1502217k

Cited by 20 publications

(12 citation statements)

References 29 publications

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“…Kara defined the Fibonacci difference matrix

\overset{F}{true} = false({\overset{f}{true}}_{n k} false)

{\hat{f}}_{n k} = \{\begin{matrix} - \frac{f_{n + 1}}{f_{n}}, & k = n - 1 \\ \frac{f_{n}}{f_{n + 1}}, & k = n \\ 0, & 0 \leq k < n - 1 or k > n \end{matrix}

and introduced some new difference sequence spaces by using this matrix. For more applications of Fibonacci numbers in sequence spaces, one can see previous works …”

Section: Introductionmentioning

confidence: 99%

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

İlkhan

2018

Math Methods in App Sciences

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Norm of an operator T:X→Y is the best possible value of U satisfying the inequality ‖Txfalse‖Y≤U‖xfalse‖X, and lower bound for T is the value of L satisfying the inequality ‖Txfalse‖Y≥L‖xfalse‖X, where ‖.‖X and ‖.‖Y are the norms on the spaces X and Y, respectively. The main goal of this paper is to compute norms and lower bounds for some matrix operators from the weighted sequence space ℓp(w) into a new space called as Fibonacci weighted difference sequence space. For this purpose, we firstly introduce the Fibonacci difference matrix trueF˜ and the space consisting of sequences whose trueF˜‐transforms are in ℓpfalse(truew˜false).

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“…Kara defined the Fibonacci difference matrix

\overset{F}{true} = false({\overset{f}{true}}_{n k} false)

{\hat{f}}_{n k} = \{\begin{matrix} - \frac{f_{n + 1}}{f_{n}}, & k = n - 1 \\ \frac{f_{n}}{f_{n + 1}}, & k = n \\ 0, & 0 \leq k < n - 1 or k > n \end{matrix}

and introduced some new difference sequence spaces by using this matrix. For more applications of Fibonacci numbers in sequence spaces, one can see previous works …”

Section: Introductionmentioning

confidence: 99%

“…For more applications of Fibonacci numbers in sequence spaces, one can see previous works. [9][10][11][12][13][14][15][16][17][18][19][20] As an application of Fibonacci numbers into infinite regular matrices, Kara and Başarır 21 have defined a new regular matrix F = (f nk ) by using the Fibonacci numbers as follows:…”

Section: Introductionmentioning

confidence: 99%

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

İlkhan

2018

Math Methods in App Sciences

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“…Since then many authors studied and generalized Fibonacci difference sequence spaces. We refer to [11,13,14,16,[30][31][32][33][34] for relevant studies. Motivated by the above studies, we introduced generalized Fibonacci difference operator by the composition of the Fibonacci band matrix F and the triple band matrix B(x, y, z).…”

Section: Fibonacci Sequence Spacesmentioning

confidence: 99%

Sequence spaces derived by the triple band generalized Fibonacci difference operator

Yaying¹,

Hazarika

Mohiuddine

et al. 2020

Adv Differ Equ

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In this article we introduce the generalized Fibonacci difference operator $\mathsf{F}(\mathsf{B})$ F ( B ) by the composition of a Fibonacci band matrix and a triple band matrix $\mathsf{B}(x,y,z)$ B ( x , y , z ) and study the spaces $\ell _{k}( \mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) . We exhibit certain topological properties, construct a Schauder basis and determine the Köthe–Toeplitz duals of the new spaces. Furthermore, we characterize certain classes of matrix mappings from the spaces $\ell _{k}(\mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) to space $\mathsf{Y}\in \{\ell _{\infty },c_{0},c,\ell _{1},cs_{0},cs,bs\}$ Y ∈ { ℓ ∞ , c 0 , c , ℓ 1 , c s 0 , c s , b s } and obtain the necessary and sufficient condition for a matrix operator to be compact from the spaces $\ell _{k}(\mathsf{F}(\mathsf{B}))$ ℓ k ( F ( B ) ) and $\ell _{\infty }(\mathsf{F}(\mathsf{B}))$ ℓ ∞ ( F ( B ) ) to $\mathsf{Y}\in \{ \ell _{\infty }, c, c_{0}, \ell _{1},cs_{0},cs,bs\} $ Y ∈ { ℓ ∞ , c , c 0 , ℓ 1 , c s 0 , c s , b s } using the Hausdorff measure of non-compactness.

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“…For further details on the normed spaces derived by the Fibonacci matrix, we refer the readers to the recent papers [4,6,[8][9][10].…”

Section: Corollary 12 Let W = (W N ) ∞mentioning

confidence: 99%

Transpose of Nörlund matrices on the domain of summability matrices

Talebi

2019

J Inequal Appl

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Compactness of matrix operators on some sequence spaces derived by Fibonacci numbers

Abstract: Abstract. In this paper, we apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for certain matrix operators on the Fibonacci difference sequence spaces p ( F ) and ∞ ( F ) to be compact, where 1 ≤ p < ∞.

Cited by 20 publications

References 29 publications

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

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

Sequence spaces derived by the triple band generalized Fibonacci difference operator

Transpose of Nörlund matrices on the domain of summability matrices

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