Some properties of pre-quasi norm on Orlicz sequence space

In this article, we introduce the concept of pre-quasi norm on E (Orlicz sequence space), which is more general than the usual norm, and give the conditions on E equipped with the pre-quasi norm to be Banach space. We give the necessity and sufficient conditions on E equipped with the pre-quasi norm such that the multiplication operator defined on E is a bounded, approximable, invertible, Fredholm, and closed range operator. The components of pre-quasi operator ideal formed by the sequence of s-numbers and E is strictly contained for different Orlicz functions are determined. Furthermore, we give the sufficient conditions on E equipped with a pre-modular such that the pre-quasi Banach operator ideal constructed by s-numbers and E is simple and its components are closed. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and E is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to E.


Introduction
Throughout the paper, we denote the space of all bounded linear operators from a Banach space X into a Banach space Y by L(X, Y ), and if X = Y , we write L(X), the space of all real sequences is denoted by w, the real numbers R, the complex numbers C, N = {0, 1, 2, . . .}, the space of null sequences by C 0 , and the space of bounded sequences by ∞ . In operator theory, the multiplication operators on L p -spaces are related to the composition operators; this means that the properties of composition operators on L p -spaces can be stated by the properties of multiplication operators. Singh and Kumar [28] proved that a composition operator on L p (X; C) is compact if and only if the multiplication operator T α is compact, where α = dμT -1 dμ is the Radon-Nikodym derivative of the measure μT -1 with respect to the measure μ. In the theory of Hilbert space, every normal operator on a separable Hilbert space is unitarily equivalent to a multiplication operator. In the spectral theory, multiplication operators have their roots in the spectral theory. For more details on multiplication operators, see [1,26,27,[29][30][31]. On sequence spaces, Mursaleen and Noman in [17,18] studied the compact operators on some difference sequence spaces; Komal and Gupta [10] studied the multiplication operators on Orlicz spaces equipped with the Luxemburg norm, and Komal et al. [11] examined the multiplication operators on Cesáro sequence spaces. The theory of operator ideal goals possesses an uncommon essentialness in useful examination. Some of operator ideals in the class of Banach spaces or Hilbert spaces are defined by different scalar sequence spaces. For example the ideal of compact operators is defined by the space C 0 of null sequence and Kolmogorov numbers. Pietsch [24] examined the quasi-ideals formed by the approximation numbers and classical sequence space p (0 < p < ∞). He proved that the ideals of nuclear operators and of Hilbert-Schmidt operators between Hilbert spaces are defined by 1 and 2 , respectively. He proved that the class of all finite rank operators is dense in the Banach quasi-ideal, and the algebra L( p ), where (1 ≤ p < ∞), contains one and only one nontrivial closed ideal. Pietsch [23] showed that the quasi Banach operator ideal formed by the sequence of approximation numbers is small. Makarov and Faried [14] proved that the quasi-operator ideal formed by the sequence of approximation numbers is strictly contained for different powers, i.e., for any infinite dimensional Banach spaces X, Y and for any q > p > 0, it is true that S In [8], Faried and Bakery studied the operator ideals constructed by approximation numbers, generalized Cesáro and Orlicz sequence spaces M . In [9], Faried and Bakery introduced the concept of pre-quasi operator ideal, which is more general than the usual classes of operator ideals. They studied the operator ideals constructed by s-numbers, generalized Cesáro and Orlicz sequence spaces M , and proved that the operator ideal formed by the previous sequence spaces and approximation numbers is small under certain conditions. The aim of this paper to study the concept of pre-quasi norm on E (Orlicz sequence space), which is more general than the usual norm, and give the conditions for E equipped with the pre-quasi norm to be Banach space. We give the necessity and sufficient conditions on E equipped with the pre-quasi norm such that the multiplication operator defined on E is a bounded, approximable, invertible, Fredholm, and closed range operator. The components of pre-quasi operator ideal formed by the sequence of s-numbers and E is strictly contained for different Orlicz functions are determined. Furthermore, we give the sufficient conditions on E equipped with a premodular such that the pre-quasi Banach operator ideal constructed by s-numbers and E is simple and its components are closed. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and E is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to E.

Definition 2.5 ([24]) A bounded linear operator
has compact closure, where B 1 denotes the closed unit ball of E. The space of all compact operators on E is denoted by L c (E).   Lindenstrauss and Tzafriri [13] utilized the idea of an Olicz function to define Orlicz sequence space: is a Banach space with the Luxemburg norm: Every Orlicz sequence space contains a subspace that is isomorphic to c 0 or q for some 1 ≤ q < ∞. As of late, different classes of sequences have been studied using Orlicz functions by Et et al. [7], Mursaleen et al. [19][20][21], Alotaibi et al. [2][3][4], and Mohiuddine et al. [15]. (ii) There exists L ≥ 1 such that (βu) ≤ L|β| (u) for all u ∈ E and for any scalar β;

Definition 2.10 ([6]) A class of linear sequence spaces
The set of all finite sequences is -dense in E. This means that, for each Closed ideal means an ideal which contains its limit points. The concept of pre-quasi operator ideal is more general than the usual classes of operator ideals.
is said to be a pre-quasi norm on the ideal Ω if the following conditions hold: (1) For all T ∈ Ω(X, Y ), g(T) ≥ 0 and g(T) = 0 if and only if T = 0; (2) There exists a constant M ≥ 1 such that g(λT) ≤ M|λ|g(T) for all T ∈ Ω(X, Y ) and

Definition 2.17 ([25]
) An s-number function is a map defined on L(X, Y ) which associates with each operator T ∈ L(X, Y ) a nonnegative scaler sequence (s n (T)) ∞ n=0 assuming that the taking after states are verified: , where X 0 and Y 0 are arbitrary Banach spaces; (d) If G ∈ L(X, Y ) and λ ∈ R, we obtain s n (λG) = |λ|s n (G); (e) Rank property: If rank(T) ≤ n, then s n (T) = 0 for each T ∈ L(X, Y ); (f ) Norming property: s r≥n (I n ) = 0 or s r<n (I n ) = 1, where I n represents the unit operator on the n-dimensional Hilbert space n 2 .

Main results
In this part, we give the concept of pre-quasi norm on Orlicz sequence space, which is more general than the usual norm, and give the conditions for Orlicz sequence space equipped with the pre-quasi norm to be a Banach space.
for all x ∈ E and for any scalar λ; The space E with is called pre-quasi normed (sss) and is denoted by E , which gives a class more general than the quasi normed space. If the space E is complete with , then E is called a pre-quasi Banach (sss). (ii) Assume λ ∈ R, x ∈ l M , and since M satisfies 2 -condition, we get a number a > 0 such that where L = max{1, a}.
(iii) Let x, y ∈ l M . Since M is nondecreasing, convex, and satisfying 2 -condition, then there exists a number a > 0 such that M |y n | = K (x) + (y) for some K = max{1, a 2 }. Hence ( M ) is a pre-quasi normed (sss). Since M is continuous and nondecreasing, hence M -1 exists. To prove that ( M ) is a pre-quasi Banach (sss), suppose x n = (x n k ) ∞ k=0 to be a Cauchy sequence in ( M ) , then for every ε > 0, there exists a natural number n 0 ∈ N such that, for all n, m ≥ n 0 , one has So (x m k ) is a Cauchy sequence in R for fixed k ∈ N, this gives lim m→∞ x m k = x 0 k for fixed k ∈ N. Hence (x nx 0 ) < ε. Finally, to prove that x 0 ∈ M , we have This means that ( M ) is a pre-quasi Banach (sss).

Multiplication operator on pre-quasi normed (sss)
In this part, we define a multiplication operator on Orlicz sequence space with a pre-quasi norm and give the necessity and sufficient conditions on Orlicz sequence space equipped with the pre-quasi norm such that the multiplication operator defined on Orlicz sequence space is a bounded, approximable, invertible, Fredholm, and closed range operator. Definition 4.1 Let α : N → C be a bounded sequence and E be a pre-quasi normed (sss), the multiplication operator is defined as for all x ∈ E. If T α is continuous, we call it a multiplication operator induced by α.

Theorem 4.2 If α : N → C is a mapping and M is an Orlicz function satisfying 2condition, then α ∈ ∞ if and only if T
Proof Let α ∈ ∞ . Then there exists C > 0 such that |α n | ≤ C for all n ∈ N. For x ∈ ( M ) , since M is nondecreasing and satisfying 2 -condition, we have where D is a constant depending on C, which implies that T α ∈ L(( M ) ).
Conversely, suppose that T α ∈ L(( M ) ). We prove that α ∈ ∞ . For, if α is not a bounded function, then for every n ∈ N, there exists some i n ∈ N such that α i n > n. Since M is nondecreasing, we obtain This proves that T α is not a bounded operator. Hence, α must be a bounded function. Proof Let |α n | = 1 for all n ∈ N. Then for all x ∈ ( M ) . Hence T α is an isometry. Conversely, suppose that |α n | < 1 for some n = n 0 . Since M is nondecreasing, we have Similarly, if |α n 0 | > 1, then we can show that (T α e n 0 ) > (e n 0 ). In both cases, we get contradiction. Hence, |α n | = 1 for all n ∈ N. Proof Suppose that T α is an approximable operator, hence T α is a compact operator. We show that lim n→∞ α n = 0. For if this were not true, then there exists δ > 0 such that the set for all d n , d m ∈ B δ . This proves that {e d n : d n ∈ B δ } is a bounded sequence which cannot have a convergent subsequence under T α . This shows that T α cannot be compact, hence it is not an approximable operator, which is a contradiction. Hence, lim n→∞ α n = 0. Conversely, suppose lim n→∞ α n = 0. Then, for every δ > 0, the set B δ = {n ∈ N : |α n | ≥ δ} is a finite set. Then is a finite dimensional space for each δ > 0. Therefore, T α |(( M ) ) B δ is a finite rank operator. For each n ∈ N, define α n : N → C by Clearly, T α n is a finite rank operator as the space (( M ) ) B 1 n is finite dimensional for each n ∈ N. Now, since M is convex and nondecreasing, we have This proves that T α -T α n ≤ 1 n and that T α is a limit of finite rank operators and hence, T α is an approximable operator.
Proof It is easy, so omitted.  Proof Suppose that α is bounded away from zero on ker(α) c . Then there exists > 0 such that |α n | ≥ for all n ∈ ker(α) c . We have to prove that range (T α ) is closed. Let z be a limit point of range (T α ). Then there exists a sequence T α x n in ( M ) , for all n ∈ N such that lim n→∞ T α x n = z. Clearly, the sequence T α x n is a Cauchy sequence. Now, since M is nondecreasing, we have This proves that {y n } is a Cauchy sequence in ( M ) . But ( M ) is complete. Therefore, there exists x ∈ ( M ) such that lim n→∞ y n = x. In view of the continuity of T α , lim n→∞ T α y n = T α x. But lim n→∞ T α x n = lim n→∞ T α y n = z. Therefore, T α x = z. Hence z ∈ range(T α ). This proves that T α has closed range. Conversely, suppose that T α has closed range. Then T α is bounded away from zero on (( M ) ) ker(α) c . That is, there exists > 0 such that (T α x) ≥  Proof Suppose that the condition is true. Define β : N → C by β n = 1 α n . Then T α and T β are bounded linear operators in view of Theorem 4.2. Also T α .T β = T β .T α = I. Hence, T β is the inverse of T α . Conversely, suppose that T α is invertible. Then range(T α ) = (( M ) ) N . Therefore, range(T α ) is closed. Hence, by Theorem 4.7, there exists a > 0 such that |α n | ≥ a for all n ∈ ker(α) c . Now ker(α) = φ; otherwise α n 0 = 0 for some n 0 ∈ N, in which case e n 0 ∈ ker(T α ), which is a contradiction, since ker(T α ) is trivial. Hence, |α n | ≥ a for all n ∈ N. Since T α is bounded, so by Theorem 4.2, there exists A > 0 such that |α n | ≤ A for all n ∈ N. Thus, we have proved that a ≤ |α n | ≤ A for all n ∈ N. (ii) |α n | ≥ for all n ∈ ker(α) c .
Proof Suppose that T α is Fredholm. If ker(α) is an infinite subset of N, then e n ∈ ker(T α ) for all n ∈ ker(α). But e n s are linearly independent, which shows that ker(T α ) is infinite dimensional, which is a contradiction. Hence, ker(α) must be a finite subset of N. Condition (ii) follows from Theorem 4.7. Conversely, if conditions (i) and (ii) are true, then we prove that T α is Fredholm. In view of Theorem 4.7, condition (ii) implies that T α has closed range. Condition (i) implies that ker(T α ) and (range(T α )) c are finite dimensional. This proves that T α is Fredholm. we have (s n (T)) ∞ n=0 ∈ ( M ) ρ , then T ∈ S ( M ) ρ (X, Y ).

Pre-quasi simple Banach operator ideal
We give here the sufficient conditions on Orlicz sequence space such that the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space are strictly contained for different Orlicz functions.
It is easy to verify that S ϕ 2 (X, Y ) ⊂ L(X, Y ). Next, if we take (s n (T)) ∞ n=0 such that ϕ 2 (s n (T)) = 1 n+1 , one can find T ∈ L(X, Y ) such that T does not belong to S ϕ 2 (X, Y ). This completes the proof. Corollary 6.2 For any infinite dimensional Banach spaces X, Y and 0 < p < q < ∞, then S p (X, Y ) S q (X, Y ) L(X, Y ).
Proof Suppose that there exists T ∈ L(S ϕ 2 , S ϕ 1 ) which is not approximable. According to Lemma 2.3, we can find X ∈ L(S ϕ 2 , S ϕ 2 ) and B ∈ L(S ϕ 1 , S ϕ 1 ) with BTXI k = I k . Then it follows for all k ∈ N that I k S ϕ 1 = ∞ n=0 ϕ 1 s n (I k ) ≤ BTX I k S ϕ 2 ≤ ∞ n=0 ϕ 2 s n (I k ) .