Statistical matrix summability of difference sequences with speed and characterization of matrix classes
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Statistical matrix summability of difference sequences with speed and characterization of matrix classes Shilpa Das1 · Hemen Dutta1 Received: 20 December 2019 / Accepted: 5 May 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract The paper introduces the notion of A-statistical convergence of difference sequences with speed λ = (λk), where 0 0, where } { K𝜖 = k ∶ ||𝜉k − 𝜉 || ≥ 𝜖 .
By Kolk [9]( (also ) see [11]) a sequence x is said to be A - statistically convergent to a number 𝜉 if 𝛿A K𝜖 = 0 for every 𝜖 > 0 . In this case, we write
stA − lim𝜉k = 𝜉. k
( ) A sequence x =( 𝜉k) is said to be statistically ∆-convergent or ∆-statistically convergent ∕ to a number 𝜉 if 𝛿 K𝜖 = 0 for every 𝜖 > 0, where } { K𝜖∕ = k ∶ ||Δ𝜉k − 𝜉 || ≥ 𝜖 .
A sequence x is(said) to be A - statistically ∆-convergent or A-∆-statistically convergent ∕ to a number 𝜉 if 𝛿A K𝜖 = 0 for every 𝜖 > 0 . In this case, we write
stA − limΔ𝜉k = 𝜉. k
The set of all A-statistically convergent sequences is denoted by stA , the set of all sequences converging A-statistically to 0 by st0A , the set of all A-statistically ∆-convergent sequences by stAΔ , and the set of all sequences converging A-∆-statistically to 0 by st0A , Δ respectively. It is observed that every convergent sequence is A-statistically convergent and every ∆-convergent sequence is A-statistically ∆-convergent. If we consider A as the identity matrix, then the concept of A-statistical convergence coincides with the concept of
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S. Das, H. Dutta
ordinary convergence, the concept of A-statistical ∆-convergence coincides with the concept of ∆-convergence, and if we consider A as the Cesàro matrix, then coincide with the concepts of statistical convergence and statistical ∆-convergence, respectively. Let λ = (𝜆k ) be a sequence with ( ) 0 k0 and n∈ℕ. We assume λ = (𝜆k ) to be a monotonically increasing positive sequence in the following Theorems. −1 Theorem 2.1 ([12]). Let X be a section-closed ) that contains 𝜆 ( and Y an) ( sequence space arbitrary sequence space. Then a matrix B ∈ z − st𝜆A ∩ X, Y if and only if B ∈ z𝜆 ∩ X, Y and the relation ( ) B[K] ∈ (X, Y) 𝛿A (K) = 0 (2.1)
is satisfied.
Theorem 2.2 ([12]). Let X be a (section-closed )sequence space and ) ( Y an arbitrary sequence space. Then a matrix B ∈ n − st𝜆A ∩ X, Y if and only if B ∈ n𝜆 ∩ X, Y and the relation (2.1) is satisfied. The following Lemma is similar to that of Lemma 2.1, and can be obtained similarly.
( ) Lemma 2.3 A sequence x = 𝜉k converges A-∆-statistically to 𝜉 if and only if there exists an infinite set of indices K = (ki ) so that the subsequence (𝜉ki ) is ∆-convergent to 𝜉 and 𝛿A (ℕ ⧵ K) = 0 and hence 𝛿A (K) = 1. Theorem 2.3 A sequence x = (𝜉k ) converges A𝜆Δ-statistically to 𝜉 if and only if there exists an infinite set of indices K = (ki ) so that the subsequence (𝜉ki ) is 𝜆-∆-convergent to 𝜉 and 𝛿A (ℕ ⧵ K) = 0 and hence 𝛿A (K) = 1. Proof Since for a bounded sequence𝜆 , we havestA𝜆 = stAΔ , so in this case Theorem 2.3 Δ coincides with
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