The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces

Definition 1.1

[1] An Orlicz function is a function φ : [ 0 , ∞ ) → [ 0 , ∞ ) , which is non-decreasing, convex and continuous with φ ( 0 ) = 0 , φ ( x ) > 0 for x > 0 and lim x → ∞ φ ( x ) = ∞ .

Definition 1.2

[2] An Orlicz function φ is said to fulfill Δ 2 -condition, for each value of x ≥ 0 , if there is a > 0 , such that φ ( 2 x ) ≤ a φ ( x ) . The Δ 2 -condition is compared to φ ( m x ) ≤ a m φ ( x ) , for all values of m > 1 and x.

Definition 1.3

[19] The Matuszewska-Orlicz lower index α φ of an Orlicz function φ is defined as follows

Theorem 1.4

[19] For any Orlicz function φ , we have α φ > 1 if and only if ℓ φ ⊂ c e s φ . Specifically, if α φ > 1 , then c e s φ ≠ { 0 } .

Theorem 1.5

[19] Let φ 1 and φ 2 be two Orlicz functions. If there exist b , t 0 > 0 such that φ 2 ( t 0 ) > 0 and φ 2 ( t ) ≤ φ 1 ( b t ) , for every t ∈ [ 0 , t 0 ] , then c e s φ 1 ⊂ c e s φ 2 .

Theorem 1.6

[19] Let φ 1 and φ 2 be two Orlicz functions and α φ 1 > 1 , then c e s φ 1 ⊂ c e s φ 2 if and only if there exist b , t 0 > 0 such that φ 2 ( t 0 ) > 0 and φ 2 ( t ) ≤ φ 1 ( b t ) , for all t ∈ [ 0 , t 0 ] .

Definition 2.1

[30] A finite rank operator is a bounded linear operator whose dimension of the range space is finite. The space of all finite rank operators on E is denoted by F ( E ) .

Definition 2.2

[30] A bounded linear operator A : E → E (where E is a Banach space) is called approximable if there are S n ∈ F ( E ) , for all n ∈ ℕ such that lim n → ∞ ∥ A − S n ∥ = 0 . The space of all approximable operators on E is denoted by Ψ ( E ) , and the space of all approximable operators from E to F is denoted by Ψ ( E , F ) .

Lemma 2.3

[30] Let T ∈ L ( X , Y ) . If T is not approximable, then there are operators G ∈ L ( X ) and B ∈ L ( Y ) such that B T G e k = e k , for all k ∈ ℕ .

Definition 2.4

[30] A Banach space E is called simple if the algebra L ( E ) contains one and only one non-trivial closed ideal.

Definition 2.5

[30] A bounded linear operator A : E → E (where E is a Banach space) is called compact if A ( B ) has compact closure, where B denotes the closed unit ball of E. The space of all compact operators on E is denoted by L c ( E ) .

Theorem 2.6

[30] If E is an infinite dimensional Banach space, we have

Definition 2.7

[42] A bounded linear operator A : E → E is called Fredholm if A has closed range, dim ( ker A ) and co-dim(range A) are finite.

Definition 2.8

[34] A class of linear sequence spaces E is called a special space of sequences (sss) if

1. e i ∈ E , for all i ∈ ℕ ,

2. if u = ( u i ) ∈ w , v = ( v i ) ∈ E and | u i | ≤ | v i | , for every i ∈ ℕ , then u ∈ E “i.e. E is solid,”

3. if ( u i ) i = 0 ∞ ∈ E , then ( u [ i 2 ] ) i = 0 ∞ ∈ E , wherever [ i 2 ] means the integral part of i 2 .

Theorem 2.9

[33] If φ is an Orlicz function satisfying Δ 2 -condition and α φ > 1 , then c e s φ is (sss).

Definition 2.10

[33] An operator ideal

is called a pre-quasi operator ideal, if there is a function g : Ω → [ 0 , ∞ ) satisfying the following conditions:

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 ( λ T ) ≤ M | λ | g ( T ) , for all T ∈ Ω ( X , Y ) and λ ∈ ℝ ,

3. there exists a constant K ≥ 1 such that g ( T 1 + T 2 ) ≤ K [ g ( T 1 ) + g ( T 2 ) ] , for all T 1 , T 2 ∈ Ω ( X , Y ) ,

4. there exists a constant C ≥ 1 such that if T ∈ L ( X 0 , X ) , P ∈ Ω ( X , Y ) and R ∈ L ( Y , Y 0 ) , then g ( R P T ) ≤ C ∥ R ∥ g ( P ) ∥ T ∥ , where X 0 and Y 0 are normed spaces.

Definition 2.11

[43] An s-number function is a map defined on L ( X , Y ) , which associates with each operator T ∈ L ( X , Y ) a non-negative scalar sequence ( s n ( T ) ) n = 0 ∞ assuming that the taking after states are verified:

1. ∥ T ∥ = s 0 ( T ) ≥ s 1 ( T ) ≥ s 2 ( T ) ≥ … ≥ 0 , for T ∈ L ( X , Y ) ,

2. s m + n − 1 ( T 1 + T 2 ) ≤ s m ( T 1 ) + s n ( T 2 ) , for all T 1 , T 2 ∈ L ( X , Y ) , m, n ∈ ℕ ,

3. ideal property: s n ( R V T ) ≤ ∥ R ∥ s n ( V ) ∥ T ∥ , for all T ∈ L ( X 0 , X ) , V ∈ L ( X , Y ) and R ∈ L ( Y , Y 0 ) , where X 0 and Y 0 are arbitrary Banach spaces,

4. if G ∈ L ( X , Y ) and λ ∈ ℝ , we obtain s n ( λ G ) = | λ | s n ( G ) ,

5. rank property: if rank ( T ) ≤ n , then s n ( T ) = 0 for each T ∈ L ( X , Y ) and

6. 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 ℓ 2 n .

Notations 2.12

[34]

Theorem 2.13

[33] The function g ( P ) = ∑ i = 0 ∞ φ ∑ j = 0 i | s j ( P ) | i + 1 is a pre-quasi norm on S c e s φ , if φ is an Orlicz function satisfying Δ 2 -condition and α φ > 1 .

Theorem 2.14

[33] If X and Y are Banach spaces, φ is an Orlicz function satisfying Δ 2 -condition and α φ > 1 , then ( S c e s φ ( X , Y ) , g ) is a pre-quasi Banach operator ideal.

Definition 3.1

Let E be (sss). If there is a function ϱ : E → [ 0 , ∞ ) fulfilling the following conditions:

1. ϱ ( x ) ≥ 0 , for every x ∈ E and ϱ ( x ) = 0 ⇔ x = θ , where θ is the zero element of E ,

2. there exists L ≥ 1 such that ϱ ( λ x ) ≤ L | λ | ϱ ( x ) , for all x ∈ E , and for any scalar λ ,

3. for some K ≥ 1 , we have ϱ ( x + y ) ≤ K ( ϱ ( x ) + ϱ ( y ) ) for every x , y ∈ E .

Theorem 3.2

Every quasi-norm is pre-quasi norm.

Theorem 3.3

If φ is an Orlicz function satisfying Δ 2 -condition and α φ > 1 , then ( c e s φ ) ϱ is a pre-quasi Banach (sss).

Proof

The function ϱ : c e s φ → [ 0 , ∞ ) , where ϱ ( x ) = ∑ n = 0 ∞ φ ∑ k = 0 n | x k | n + 1 , for all x ∈ c e s φ , is satisfying the following conditions:

1. ϱ ( x ) ≥ 0 , for each x ∈ c e s φ and ϱ ( x ) = 0 ⇔ x = θ .

2. Suppose λ ∈ ℝ , x ∈ c e s φ and since φ is satisfying Δ 2 -condition and α φ > 1 , we have a number b > 0 such that

   ϱ ( λ x ) = ∑ n = 0 ∞ φ ∑ k = 0 n | λ x k | n + 1 ≤ | λ | b ∑ n = 0 ∞ φ ∑ k = 0 n | x k | n + 1 = L | λ | ϱ ( x ) ,

   where L = max { 1 , b } .

3. Let x , y ∈ c e s φ . Since φ is non-decreasing, convex and satisfying Δ 2 -condition and α φ > 1 , then there exists a number b > 0 such that

for some K = max { 1 , b 2 } . Hence, ( c e s φ ) ϱ is a pre-quasi normed (sss). Since φ is continuous and non-decreasing, hence φ − 1 exists, to prove that ( c e s φ ) ϱ is a pre-quasi Banach (sss), let x n = ( x k n ) k = 0 ∞ be a Cauchy sequence in ( c e s φ ) ϱ , then for every ε ∈ ( 0 , 1 ) , there exists n 0 ∈ ℕ such that, for all n , m ≥ n 0 , one has

So, ( x i m ) is a Cauchy sequence in ℝ for fixed i ∈ ℕ , this gives lim i → ∞ x i m = x i 0 for fixed i ∈ ℕ . Hence ϱ ( x n − x 0 ) < ε . Finally, to prove that x 0 ∈ c e s φ , we have

so x 0 ∈ c e s φ . This means that ( c e s φ ) ϱ is a pre-quasi Banach (sss).□

Corollary 3.4

( c e s p ) ϱ , where ϱ ( x ) = ∑ n = 0 ∞ ∑ k = 0 n | x k | n + 1 p , for all x ∈ c e s p , is a pre-quasi Banach (sss), if 1 < p < ∞ .

Definition 4.1

Let α = ( α k ) be a sequence of complex numbers and X ϱ be a pre-quasi normed (sss). The multiplication operator on X is defined as T α : X → X , T α x = ( α k x k ) , for all x ∈ X . If the multiplication operator T α is continuous, then it is called a multiplication operator induced by α .

Theorem 4.2

If α ∈ ℂ ℕ , φ is an Orlicz function satisfying Δ 2 -condition and α φ > 1 , then α ∈ ℓ ∞ if and only if T α ∈ L ( ( c e s φ ) ϱ ) .

Proof

Let α ∈ ℓ ∞ . Then, there exists C > 0 such that | α n | ≤ C , for all n ∈ ℕ . For x ∈ ( c e s φ ) ϱ , since φ is an Orlicz function satisfying Δ 2 -condition and α φ > 1 , we have

where D is a constant that depends on C, which gives that T α ∈ L ( ( c e s φ ) ϱ ) .

Conversely, suppose that T α ∈ L ( ( c e s φ ) ϱ ) . We prove that α ∈ ℓ ∞ . If α ∉ ℓ ∞ , then for every n ∈ ℕ , there exists some i n ∈ ℕ such that α i n > n . Since φ is non-decreasing and continuous, we obtain

This proves that T α is not a bounded operator. Hence, α must be a bounded function.□

Theorem 4.3

Let α ∈ ℂ ℕ and ( c e s φ ) ϱ be a pre-quasi normed (sss), then | α n | = 1 , for all n ∈ ℕ if and only if T α is an isometry.

Proof

Let | α n | = 1 , for all n ∈ ℕ . Then,

for all x ∈ ( c e s φ ) ϱ . Hence, T α is an isometry.

Conversely, suppose that | α i | < 1 for some i = i 0 . Since φ is non-decreasing, we have

Similarly, if | α n 0 | > 1 , then we can show that ϱ ( T α e n 0 ) > ϱ ( e n 0 ) . In the two cases, we obtain contradiction. Hence, | α n | = 1 , for each n ∈ ℕ .□

Theorem 4.4

Let α ∈ ℂ ℕ , φ be an Orlicz function satisfying Δ 2 -condition, α φ > 1 and T α ∈ L ( ( c e s φ ) ϱ ) . Then, T α ∈ Ψ ( ( c e s φ ) ϱ ) if and only if ( α n ) n = 0 ∞ ∈ c 0 .

Proof

Suppose T α is an approximable operator, hence T α is a compact operator. We show that lim n → ∞ α n = 0 . If this were not valid, then there exists δ > 0 such that the set B δ = { r ∈ ℕ : | α r | ≥ δ } is an infinite set. Let d 1 , d 2 , … , d n , … be in B δ . Then, { e d i : d i ∈ B δ } is an infinite bounded set in ( c e s φ ) ϱ . Consider

for all d i , d j ∈ B δ . This proves { e d i : d i ∈ B δ } is a bounded sequence which cannot have a convergent subsequence under T α . This shows that T α cannot be a compact, hence is not 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, ( ( c e s φ ) ϱ ) B δ = { x = ( x n ) ∈ ℂ ℕ : x n = x n , if n ∈ B δ or x n = 0 , otherwise } is a finite dimensional space for each δ > 0 . Therefore, T α | ( ( c e s φ ) ϱ ) B δ is a finite rank operator. For each n ∈ ℕ , define α n ∈ ℂ ℕ by

Clearly, T α n is a finite rank operator as the space ( ( c e s φ ) ϱ ) B 1 n is finite dimensional for each n ∈ ℕ . Now, since φ is convex and non-decreasing, 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.□

Theorem 4.5

Let α ∈ ℂ ℕ , φ be an Orlicz function satisfying Δ 2 -condition, α φ > 1 and T α ∈ L ( ( c e s φ ) ϱ ) . Then, T α ∈ L c ( ( c e s φ ) ϱ ) if and only if ( α n ) n = 0 ∞ ∈ c 0 .

Proof

It is easy so omitted.□

Corollary 4.6

Let φ be an Orlicz function satisfying Δ 2 -condition and α φ > 1 , we have L c ( ( c e s φ ) ϱ ) ⫋ L ( ( c e s φ ) ϱ ) .

Proof

Since the identity operator I on ( c e s φ ) ϱ is a multiplication operator induced by the sequence α = ( 1 , 1 , … ) , hence I ∉ L c ( ( c e s φ ) ϱ ) and I ∈ L ( ( c e s φ ) ϱ ) .□

Theorem 4.7

Let ( c e s φ ) ϱ be a pre-quasi Banach (sss) and T α ∈ L ( ( c e s φ ) ϱ ) . Then, there exist a > 0 and A > 0 such that a < | α n | < A , for all n ∈ ℕ \ ker ( α ) ≔ ker ( α ) c if and only if T α has closed range.

Proof

Suppose α is bounded away from zero on k e r ( α ) c . Then, there exists ε > 0 such that | α n | ≥ ε , for all n ∈ ker ( α ) c . We have to prove that range ( T α ) is closed. Let h be a limit point of range ( T α ) . Then, there exists a sequence T α x n in ( c e s φ ) ϱ , for all n ∈ ℕ such that lim n → ∞ T α x n = h . Clearly, the sequence T α x n is a Cauchy sequence. Now, since φ is non-decreasing, we have

where

This proves that { y n } is a Cauchy sequence in ( c e s φ ) ϱ . But ( c e s φ ) ϱ is complete. Therefore, there exists x ∈ ( c e s φ ) ϱ such that lim n → ∞ y n = x . In view of continuity of T α , hence lim n → ∞ T α y n = T α x . But lim n → ∞ T α x n = lim n → ∞ T α y n = h . Therefore, T α x = h . Hence, h ∈ 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 ( ( c e s φ ) ϱ ) ker ( α ) c . That is, there exists ε > 0 such that ϱ ( T α x ) ≥ ϱ ( ε x ) , for all x ∈ ( ( c e s φ ) ϱ ) ker ( α ) c . Let D = { k ∈ ker ( α ) c : | α k | < ε } . If D ≠ ϕ , then for n 0 ∈ D , we have