On I-convergent sequence spaces defined by a compact operator and a modulus function

ABOUT THE AUTHORS Vakeel A. Khan received his MPhil and PhD degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently, he is a senior assistant professor in the same university. He is a vigorous researcher in the area of Sequence Spaces , and has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylor’s and Francis), Information Sciences (Elsevier), Applied Mathematics A Journal of Chinese Universities and SpringerVerlag (China), Rendiconti del Circolo Matematico di Palermo Springer-Verlag (Italy), Studia Mathematica (Poland), Filomat (Serbia), Applied Mathematics & Information Sciences (USA). Mohd Shafiq did his MSc in Mathematics from University of Jammu, Jammu and Kashmir, India. Currently, he is a PhD scholar at Aligarh Muslim University, Aligarh, India. His research interests are Functional Analysis, sequence spaces, I-convergence, invariant means, zweier sequences, and interval numbers theory. Rami Kamel Ahmad Rababah is a research scholar in the Department of Mathematics, Aligarh Muslim University, Aligarh, India. PUBLIC INTEREST STATEMENT The term sequence has a great role in Analysis. Sequence spaces play an important role in various fields of Real Analysis, Complex Analysis, Functional Analysis, and Topology Convergence of sequences has always remained a subject of interest to the researchers. Several new types of convergence of sequences were studied by the researchers and named them as usual convergence, uniform convergence, strong convergence,week convergence, etc. Later, the idea of statistical convergence came into existence which is the generalization of usual convergence. Statistical convergence has several applications in different fields of Mathematics like Number Theory, Trigonometric Series, Summability Theory, Probability Theory, Measure Theory, Optimization, and Approximation Theory. The notion of Ideal convergence (I-convergence) is a generalization of the statistical convergence and equally considered by the researchers for their research purposes since its inception. Received: 28 October 2014 Accepted: 11 March 2015 Published: 20 May 2015


PUBLIC INTEREST STATEMENT
The term sequence has a great role in Analysis. Sequence spaces play an important role in various fields of Real Analysis, Complex Analysis, Functional Analysis, and Topology Convergence of sequences has always remained a subject of interest to the researchers. Several new types of convergence of sequences were studied by the researchers and named them as usual convergence, uniform convergence, strong convergence,week convergence, etc. Later, the idea of statistical convergence came into existence which is the generalization of usual convergence. Statistical convergence has several applications in different fields of Mathematics like Number Theory, Trigonometric Series, Summability Theory, Probability Theory, Measure Theory, Optimization, and Approximation Theory. The notion of Ideal convergence (I-convergence) is a generalization of the statistical convergence and equally considered by the researchers for their research purposes since its inception.
The idea of modulus was introduced by Nakano (1953). Ruckle (1967Ruckle ( , 1968Ruckle ( , 1973 used the idea of a modulus function f to construct the sequence space This space is an FK-space and Ruckle (1967Ruckle ( , 1968Ruckle ( , 1973 proved that the intersection of all such X(f ) spaces is , the space of all finite sequences.
The space X(f ) is closely related to the space 1 which is an X(f ) space with f (x) = x for all real x ≥ 0. Thus Ruckle (1967Ruckle ( , 1968Ruckle ( , 1973 proved that, for any modulus f .
where Spaces of the type X(f ) are a special case of the spaces structured by Gramsch (1967). From the point of view of local convexity, spaces of the type X(f ) are quite pathological. Symmetric sequence spaces, which are locally convex have been frequently studied by Garling (1966), Köthe (1970), and Ruckle (1967Ruckle ( , 1968Ruckle ( , 1973. The sequence spaces by the use of modulus function was further investigated by Maddox (1969Maddox ( , 1986, Khan (2005Khan ( , 2006, Bhardwaj (2003), and many others.
As a generalization of usual convergence, the concept of statistical convergent was first introduced by Fast (1951) and also independently by Buck (1953) and Schoenberg (1959) for real and complex sequences. Later on, it was further investigated from sequence space point of view and linked with the Summability Theory by Fridy (1985), Šalát (1980), Tripathy (1998), Khan (2007), Khan and Sabiha (2012), Khan, Shafiq, and Rababah (2015), and many others.
Definition 1.4 A sequence x = (x k ) ∈ is said to be statistically convergent to a limit L ∈ ℂ if for every > 0, we have where vertical lines denote the cardinality of the enclosed set.

Now, we recall the following definitions:
Definition 1.5 Let ℕ be a non-empty set. Then a family of sets I ⊆ 2 ℕ (power set of ℕ) is said to be an ideal if Definition 1.6 A non-empty family of sets £(I) ⊆ 2 ℕ is said to be filter on ℕ if and only if (1) Φ ∉ £(I), In this case, we write I − lim x k = L.
Definition 1.14 A sequence x = (x k ) ∈ is said to be I-bounded if there exists some M > 0 such that {k ∈ ℕ : |x k | ≥ M} ∈ I.
where is a permutation on ℕ.
A canonical preimage of a step space E K is a set of preimages all elements in E K . i.e. y is in the canonical preimage of E K iff y is the canonical preimage of some x ∈ E K .
Definition 1.21 A sequence space E is said to be monotone if it contains the canonical preimages of its step space.
{{x} :  Khan et al., 2015;Kostyrko et al., 2000). If I = I f , the class of all finite subsets of v. Then, I is an admissible ideal in v and I f convergence coincides with the usual convergence.
Then, I is an admissible ideal in ℕ and we call the I -convergence as the logarithmic statistical convergence.
Then, I is an admissible ideal in ℕ and we call the I d -convergence as the asymptotic statistical convergence.
The convergence field F(I) is a closed linear subspace of l ∞ with respect to the supremum norm, Šalát et al., 2004, 2005).
is a sequence of scalars with n → (n → ∞) and x n , a ∈ X with x n → a (n → ∞) in the sense that g(x n − a) → 0 (n → ∞) , then g( n x n − a) → 0 (n → ∞).
The notation of paranorm sequence spaces was studied at the initial stage by Nakano (1953). Later on, it was further investigated by Maddox (1969), Tripathy and Hazarika (2009), Khan et al. (2013), and the references therein.
Throughout the article, we use the same techniques as used in Tripathy andHazarika (2009, 2011).
We used the following lemmas for establishing some results of this article.
Lemma 2 (see, Tripathy & Hazarika, 2009. represent the I-convergent, I-null, I-Bounded , bounded I-convergent, and bounded I-null Sequences spaces defined by a compact operator T on the real space ℝ, respectively.

Main results
In this article, we introduce the following classes of sequences.
We also denote Theorem 2.1 Let f be a modulus function. Then, the classes of sequences  I (f ), Proof We shall prove the result for  I (f ). The proof for the other spaces will follow similarly. For, let x = (x k ), y = (y k ) ∈  I (f ) and , be scalars. Then, for a given > 0, we have Let be such that A c 1 , A c 2 ∈ I.
Since f is a modulus function, we have Therefore, implies that A 3 ∈ £(I). Thus, A c 3 = A c 1 ∪ A c 2 ∈ I. Therefore, x k + y k ∈  I (f ), for all scalars , , and (x k ), (y k ) ∈  I (f ).
, for some L 2 ∈ ℂ ∈ (I) Hence,  I (f ) is a linear space.
in the sense that Then, since the inequality holds by subadditivity of g, the sequence {g(x k )} is bounded.
Therefore, as (k → ∞). That is to say that scalar multiplication is continuous. Hence, Suppose that L = I − lim x. Then, the set Fix an N ∈ B . Then we have, If we fix an > 0 then we have C ∈ £(I) as well as C 2 ∈ £(I).
Hence C ∩ C 2

∈ £(I). This implies that
That is That is where the diam of J denotes the length of interval J.
Theorem 2.4 Let f 1 and f 2 be two modulus functions and satisfying Δ 2 − Condition, then Proof (a) Let x = (x k ) ∈  I • (f 2 ) be any arbitrary element. Then, the set Let > 0 and choose with 0 < < 1 such that f 1 (t) < , 0 ≤ t ≤ .
Let us denote and consider Now, since f 1 is an modulus function, we have For y k > , we have Now, since f 1 is non-decreasing and modulus, it follows that Again, since f 1 satisfies Δ 2 − Condition, we have Thus, f 1 y k < K (y k ) f 1 2 Hence, Therefore, from Equations 2.11-2.13, we have ( the inclusions can be established similarly.
. Let > 0 be given. Then, the sets and Therefore, from Equations 2.14 and 2.15 the set Then the result follows from Equation 2.16 and the following inequality.
That the space  I • (f ) is monotone follows from the Lemma (I). Hence  I • (f ) is solid and monotone.
Theorem 2.7 The spaces  I (f ) and  I  (f ) are not neither solid nor monotone.
Proof Here we give a counter example for the proof of this result.
Counter example. Let I = I f and f (x) = x for all x ∈ [0, ∞). Consider the K-step  K of  defined as follows.
Let (x k ) ∈  and let (y k ) ∈  K be such that Consider the sequence (x k ) defined as by x k = 1 for all k ∈ ℕ. Then (x k ) ∈  I (f ) and  I  (f ) but its K-step preimage does not belong to  I (f ) and  I  (f ). Thus,  I (f ) and  I  (f ) are not monotone. Hence,  I (f ) and  I  (f ) are not solid by Lemma(I).
Theorem 2.8 If (x = x k ) and (y = y k ) be two sequences with T(x ⋅ y) = T(x)T(y). Then, the spaces  I (f ) and  I • (f ) are sequence algebra.
Proof Let (x = x k ) and (y = y k ) be two elements of  I Then, the sets and Therefore, Hence,  I • (f )is sequence algebra. For  I (f ), the result can be proved similarly.
Theorem 2.9 Let f be a modulus function. Then,  I Next, let (x k ) ∈  I (f ). Then there exists some L such that

We have
Taking supremum over k on both sides, we get (x k ) ∈  I ∞ (f ) Therefore, from Equations 2.19-2.21, we have for all k ∈ B x ∩ B y ∈ £(I).