A new regular infinite matrix defined by Jordan totient function and its matrix domain in ℓp

İlkhan, Merve; Şi̇mşek, Necip; Kara, Emrah Evren

doi:10.1002/mma.6501

Cited by 27 publications

(14 citation statements)

References 26 publications

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“…Recently, several authors constructed interesting Banach sequence spaces using the domain of special triangles, for instance İlkhan [26], İlkhan and Kara [24], Roopaei [54,55], Roopaei et al [53], and Yaying et al [64]. We followed this approach and introduced BK spaces k (F(B)) and ∞ (F(B)) defined as the domain of the product matrix F (B(x, y, z)) in the spaces k and ∞ , respectively.…”

Section: Resultsmentioning

confidence: 99%

“…The set X is a sequence space and is known as the domain of matrix in the space X. Additionally, if X is BK -space and is a triangle, then X is also BK -space endowed with the norm s X = s X [27], where the matrix = (ψ rv ) is called a triangle if ψ rr = 0 for all r ∈ N and ψ rv = 0 for v > r. Using this famous result several authors [4,29,35,41,48] in the literature constructed new BK -spaces. We also mention [22,23,26,[53][54][55][62][63][64] for some recent publications and textbooks [6,47,61] in this field.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

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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“…Schoenberg 23 showed that this special mapping is regular. In İlkhan et al, 21 the authors showed that the Jordan totient matrix operator is also regular.…”

Section: Some New Banach Spaces Derived By Jordan's Functionmentioning

confidence: 99%

“…The Jordan totient matrix operator

{normalΥ}^{r} = false(υ_{n k}^{r} false)

is defined by İlkhan et al 21 as

υ_{n k}^{r} = \{\begin{array}{ccl} \frac{J_{r} false(k false)}{n^{r}} & , & if k false| n \\ 0 & , & if k ∤ n \end{array}

for each

r \in ℕ

.…”

Section: Some New Banach Spaces Derived By Jordan's Functionmentioning

confidence: 99%

“…The inverse

{false({normalΥ}^{r} false)}^{- 1} = false({false(υ_{n k}^{r} false)}^{- 1} false)

of the matrix Υ r is computed in İlkhan et al 21 as

{(υ_{n k}^{r})}^{- 1} = \{\begin{array}{ccl} \frac{μ false(\frac{n}{k} false)}{J_{r} false(n false)} k^{r} & , & if k false| n \\ 0 & , & if k ∤ n . \end{array}

In this study, the sequence

y = false(y_{n} false) = ((), \frac{1}{n^{r}} \sum_{k false| n} J_{r} false(k false) x_{k})

denotes the Υ r ‐transform of a sequence x =( x k ). Further, the sequence x =( x n ) given by

x_{n} = \sum_{k false| n} \frac{μ false(\frac{n}{k} false)}{J_{r} false(n false)} k^{r} y_{k},

corresponds to the inverse transform of a sequence y =( y k ).…”

Section: Some New Banach Spaces Derived By Jordan's Functionmentioning

confidence: 99%

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Compact operators on the Jordan totient sequence spaces

İlkhan

Kara

Usta

2020

Math Methods in App Sciences

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The necessary and sufficient conditions for compactness of a matrix operator between Banach spaces is obtained by utilizing the concept of the Hausdorff measure of noncompactness. This is one of the most interesting application in the theory of sequence spaces. In this paper, the compact operators are characterized on Jordan totient sequence spaces by using the concept of the Hausdorff measure of noncompactness.

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