Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means

Awad A. Bakery; Mustafa M. Mohammed

doi:10.1515/math-2019-0135

Article Open Access

Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means

Awad A. Bakery and Mustafa M. Mohammed

Published/Copyright: December 31, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

Let E be a generalized Cesáro sequence space defined by weighted means and by using s-numbers of operators from a Banach space X into a Banach space Y. We give the sufficient (not necessary) conditions on E such that the components

SE(X,Y):={T∈L(X,Y):((sn(T))n=0∞∈E},

of the class S_E form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal S_E, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small and the pre-quasi Banach operator ideal constructed by s-numbers is simple Banach space. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to this sequence space.

Keywords: pre-quasi operator ideal; s-numbers; generalized Cesáro sequence space; weighted means; simple Banach space

MSC 2010: 46B70; 47B10; 47L20

1 Introduction

Through the paper

L(X,Y)={T:X→Y;Tis a bounded linear operator;XandYare Banach Spaces},

and if X = Y, we write L(X), by w, we denote the space of all real sequences and θ is the zero vector of E. As an aftereffect of the enormous applications in geometry of Banach spaces, spectral theory, theory of eigenvalue distributions and fixed point theorems etc., the theory of operator ideals 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 c₀ of null sequence and Kolmogorov numbers. Pietsch [1], examined the quasi-ideals formed by the approximation numbers and classical sequence space ℓ^p(0 < p < ∞). He proved that the ideals of nuclear operators and of Hilbert Schmidt operators between Hilbert spaces are defined by ℓ¹ and ℓ² respectively. He proved that the class of all finite rank operators are dense in the Banach quasi-ideal and the algebra L(ℓ^p), where (1 ≤ p < ∞) contains one and only one non-trivial closed ideal. Pietsch [2], showed that the quasi Banach Operator ideal formed by the sequence of approximation numbers is small. Makarov and Faried [3], 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 Sℓpapp(X,Y)⫋Sℓqapp(X,Y)⫋L(X,Y). In [4], Faried and Bakery studied the operator ideals constructed by approximation numbers, generalized Cesáro and Orlicz sequence spaces ℓ_M. In [5], Faried and Bakery introduced the concept of pre-quasi operator ideal which is more general than the usual classes of operator ideal, they studied the operator ideals constructed by s- numbers, generalized Cesáro and Orlicz sequence spaces ℓ_M, and proved that the operator ideal formed by the previous sequence spaces and approximation numbers is small under certain conditions. There are articles on bounded linear operators transforming between sequence spaces those have been studied by Tripathy and Paul [6], Tripathy and Saikia [7], and Tripathy and Das [8], from different aspects such as spectra, resolvent spectra etc. The aim of this paper is to study a generalized class S_E by using the sequence of s-numbers and E (generalized Cesáro sequence space defined by weighted means), we give sufficient (not necessary) conditions on E such that S_E constructs an pre-quasi operator ideal, which gives a negative answer of Rhoades [9] open problem about the linearity of E-type spaces S_E. The components of S_E as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. The pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers are determined. Finally, we show that the pre-quasi Banach operator ideal formed by E and approximation numbers is small under certain conditions. Furthermore, the sufficient conditions for which the pre-quasi Banach operator ideal constructed by s-numbers is simple Banach space. Also the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space is strictly contained in the class of all bounded linear operators whose its sequence of eigenvalues belongs to this sequence space.

2 Definitions and preliminaries

Defintion 2.1

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

∥T∥ = s₀(T) ≥ s₁(T) ≥ s₂(T) ≥ … ≥ 0, for T ∈ L(X, Y),
s_m+n−1(T₁ + T₂) ≤ s_m(T₁) + s_n(T₂) for all T₁, T₂ ∈ L(X, Y), m, n ∈ ℕ,
ideal property: s_n(RVT) ≤ ∥R∥ s_n(V) ∥T∥ for all T ∈ L(X₀, X), V ∈ L(X, Y) and R ∈ L(Y, Y₀), where X₀ and Y₀ are arbitrary Banach spaces,
if G ∈ L(X, Y) and λ ∈ ℝ, we obtain s_n(λ G) = |λ| s_n(G),
rank property: If rank(T) ≤ n, then s_n(T) = 0 for each T ∈ L(X, Y),
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 ℓ2n .

There are several examples of s-numbers, we mention the following:

The n-th approximation number, denoted by α_n(T), is defined by

α_n(T) = inf{∥T − B∥ : B ∈ L(X, Y) and rank(B) ≤ n}.
The n-th Gel’fand number, denoted by c_n(T), is defined by c_n(T) = α_n(J_Y T), where J_Y is a metric injection from the normed space Y to a higher space l_∞(Λ) for an adequate index set Λ. This number is independent of the choice of the higher space l_∞(Λ).
The n-th Kolmogorov number, denoted by d_n(T), is defined by

dn(T)=infdimY≤nsup∥x∥≤1infy∈Y∥Tx−y∥.
The n-th Weyl number, denoted by x_n(T), is defined by

xn(T)=inf{αn(TB):∥B:ℓ2→X∥≤1}.
The n-th Chang number, denoted by y_n(T), is defined by

yn(T)=inf{αn(BT):∥B:Y→ℓ2∥≤1}.
The n-th Hilbert number, denoted by h_n(T), is defined by

hn(T)=sup{αn(BTA):∥B:Y→ℓ2∥≤1and∥A:ℓ2→X∥≤1}.

Remark 2.2

[10] Among all the s-number sequences defined above, it is easy to verify that the approximation number, α_n(T) is the largest and the Hilbert number, h_n(T) is the smallest s-number sequence, i.e., h_n(T) ≤ s_n(T) ≤ α_n(T) for any bounded linear operator T. If T is compact and defined on a Hilbert space, then all the s-numbers coincide with the eigenvalues of |T|, where |T| = (T∗T)12 .

Theorem 2.3

[10, p.115] If T ∈ L(X, Y), then

hn(T)≤xn(T)≤cn(T)≤αn(T)andhn(T)≤yn(T)≤dn(T)≤αn(T).

Defintion 2.4

[1] 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).

Defintion 2.5

[1] 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 2.6

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

Defintion 2.7

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

Defintion 2.8

[1] A bounded linear operator A : E → E (where E is a Banach space) is called compact if A(B₁) has compact closure, where B₁ denotes the closed unit ball of E. The space of all compact operators on E is denoted by L_c(E).

Theorem 2.9

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

F(E)⫋Ψ(E)⫋Lc(E)⫋L(E).

Defintion 2.10

[1] Let L be the class of all bounded linear operators between any arbitrary Banach spaces. A sub class U of L is called an operator ideal if each element U(X, Y) = U ∩ L(X, Y) fulfill the following conditions:

I_ϝ ∈ U wherever ϝ represents Banach space of one dimension.
The space U(X, Y) is linear over ℝ.
If T ∈ L(X₀, X), V ∈ U(X, Y) and R ∈ L(Y, Y₀) then, RVT ∈ U(X₀, Y₀). See [11, 12].

The concept of pre-quasi operator ideal which is more general than the usual classes of operator ideal.

Defintion 2.11

[5] A function g : Ω → [0, ∞) is said to be a pre-quasi norm on the ideal Ω if the following conditions holds:

For all T ∈ Ω(X, Y), g(T) ≥ 0 and g(T) = 0 if and only if T = 0,
there exists a constant M ≥ 1 such that g(λ T) ≤ M|λ|g(T), for all T ∈ Ω(X, Y) and λ ∈ ℝ,
there exists a constant K ≥ 1 such that g(T₁ + T₂) ≤ K[g(T₁) + g(T₂)], for all T₁, T₂ ∈ Ω(X, Y),
there exists a constant C ≥ 1 such that if T ∈ L(X₀, X), P ∈ Ω(X, Y) and R ∈ L(Y, Y₀) then g(RPT) ≤ C ∥R∥ g(P) ∥T∥, where X₀ and Y₀ are normed spaces.

Theorem 2.12

[5] Every quasi norm on the ideal Ω is a pre-quasi norm on the ideal Ω.

Let (a_n), (p_n) and (q_n) be sequences of positive reals with p_n ≥ 1 for all n ∈ ℕ, Altay and Başar [13] defined the generalized Cesáro sequence space defined by weighted means as:

ces((an),(pn),(qn))={x=(x(k))∈ω:ρ(λx)<∞,for someλ>0},whereρ(x)=∑n=0∞(an∑k=0nqk|x(k)|)pn,

which is a Banach space with the Luxemburg norm defined by: ∥x∥=inf{λ>0:ρ(xλ)≤1}. If (p_n) is bounded, we can simply write

ces((an),(pn),(qn))={x=(x(k))∈ω:∑n=0∞(an∑k=0nqk|x(k)|)pn<∞}.

Remark 2.13

Taking q_n = 1 for all n ∈ ℕ, ces((a_n), (p_n), (q_n)) reduced to ces(((a_n), (p_n)), the sequence space defined and studied by Şengönül [14].
Taking an=1∑k=0nqk, then ces((a_n), (p_n), (q_n)) is reduced to ces(((p_n), (q_n)), the N̈orlund sequence spaces studied by Wang [15].
Taking a_n = 1n+1 and q_n = 1, for all n ∈ ℕ, then ces((a_n), (p_n), (q_n)) is reduced to ces((p_n)) studied by Sanhan and Suantai [16].
Taking a_n = 1n+1 , q_n = 1 and p_n = p, for all n ∈ ℕ, then ces((a_n), (p_n), (q_n)) is reduced to ces_p. Different types of Cesáro summable sequence has been studied by many authors see [17, 18, 19].

Defintion 2.14

[5] Let E be a linear space of sequences, then E is called a (sss) if:

For n ∈ ℕ, e_n ∈ E,
E is solid i.e., assuming x = (x_n) ∈ w, y = (y_n) ∈ E and |x_n| ≤ |y_n| for all n ∈ ℕ, then x ∈ E,
(x[n2])n=0∞ ∈ E, where [ n2 ] indicates the integral part of n2 , whenever (xn)n=0∞ ∈ E.

Defintion 2.15

[5] A subclass of the (sss) called a pre-modular (sss) assuming that we have a map ρ : E → [0, ∞[ with the followings:

For x ∈ E, x = θ ⇔ ρ(x) = 0 with ρ(x) ≥ 0,
for each x ∈ E and scalar λ, we get a real number L ≥ 1 for which ρ(λ x) ≤ |λ|Lρ(x),
ρ(x + y) ≤ K(ρ(x) + ρ(y)) for each x, y ∈ E, holds for a few numbers K ≥ 1,
for n ∈ ℕ, |x_n| ≤ |y_n|, we obtain ρ((x_n)) ≤ ρ((y_n)),
the inequality,

ρ((x_n)) ≤ ρ(( x[n2] )) ≤ K₀ρ((x_n)) holds, for some numbers K₀ ≥ 1,
F = E_ρ, where F is the space of finite sequences,
there is a steady ξ > 0 such that ρ(λ, 0, 0, 0, …) ≥ ξ|λ|ρ(1, 0, 0, 0, …) for any λ ∈ ℝ.

Condition (ii) gives the continuity of ρ(x) at θ. The linear space E enriched with the metric topology formed by the premodular ρ will be indicated by E_ρ. Moreover condition (1) in definition (2.14) and condition (vi) in definition (2.15) explain that (e_n)_n∈ℕ is a Schauder basis of E_ρ.

Notations 2.16

[5]

SE:={SE(X,Y);XandYare Banach Spaces},whereSE(X,Y):={T∈L(X,Y):((si(T))i=0∞∈E}.SEapp:={SEapp(X,Y);XandYare Banach Spaces},whereSEapp(X,Y):={T∈L(X,Y):((αi(T))i=0∞∈E}.

Also

SEKol:={SEKol(X,Y);XandYare Banach Spaces},whereSEKol(X,Y):={T∈L(X,Y):((di(T))i=0∞∈E}.

Theorem 2.17

[5] If E is a (sss), then S_E is an operator ideal.

Throughout, we denote e_n = {0, 0, …, 1, 0, 0, …} where 1 appears at the n^th place for all n ∈ ℕ and the given inequality will be used in the sequel:

|an+bn|pn≤H(|an|pn+|bn|pn), (1)

where H = max{1, 2^h−1}, h = sup_n p_n and p_n ≥ 1 for all n ∈ ℕ. See [13].

3 Linear problem

We examine here the operator ideals created by s-numbers also generalized Cesáro sequence space defined by weighted means such that those classes of all bounded linear operators T between arbitrary Banach spaces with (s_n(T))_n∈ℕ in this sequence space type an ideal operator.

Theorem 3.1

ces((a_n), (p_n), (q_n)) is a (sss), if the following conditions are satisfied:

The sequence (p_n) is increasing and bounded from above with p₀ > 1 for all n ∈ ℕ,
the sequence (a_n) of positive reals with ∑n=0∞ (a_n)^p_n < ∞,
either (q_n) is a monotonic decreasing sequence of positive reals or monotonic increasing bounded sequence and there exists a constant C ≥ 1 such that q_2n+1 ≤ C q_n.

Proof

(1-i) let x, y ∈ ces((a_n), (p_n), (q_n)). Since (p_n) is bounded, we have

∑n=0∞(an∑k=0nqk|xk+yk|)pn≤2h−1(∑n=0∞(an∑k=0nqk|xk|)pn+∑n=0∞(an∑k=0nqk|yk|)pn)<∞,

then x + y ∈ ces((a_n), (p_n), (q_n)).

(1-ii) let λ ∈ ℝ, x ∈ ces((a_n), (p_n), (q_n)) and since (p_n) is bounded, we have

∑n=0∞(an∑k=0nqk|λxk|)pn≤supn|λ|pn∑n=0∞(an∑k=0nqk|xk|)pn<∞.

Then λ x ∈ ces((a_n), (p_n), (q_n)), from (1-i) and (1-ii) ces((a_n), (p_n), (q_n)) is a linear space.

Also to show that e_m ∈ ces((a_n), (p_n), (q_n)), for all m ∈ ℕ. Since (p_n) with p₀ > 1 and ∑n=0∞ (a_n)^p_n < ∞, we have

∑n=0∞(an∑k=0nqk|em(k)|)pn=∑n=m∞(anqm)pn≤supn(qm)pn∑n=m∞(an)pn≤supn(qm)pn∑n=0∞(an)pn<∞.

Hence e_m ∈ ces((a_n), (p_n), (q_n)).

(2) Let |x_n| ≤ |y_n| for all n ∈ ℕ and y ∈ ces((a_n), (p_n), (q_n)). Since a_n > 0 and q_n > 0 for all n ∈ ℕ, then

∑n=0∞(an∑k=0nqk|xk|)pn≤∑n=0∞(an∑k=0nqk|yk|)pn<∞,

we get x ∈ ces((a_n), (p_n), (q_n)).

(3) Let (x_n) ∈ ces((a_n), (p_n), (q_n)), where (p_n) and (q_n) be increasing sequences of positive reals with a constant C ≥ 1 such that q_2n+1 ≤ Cq_n, then we have

∑n=0∞(an∑k=0nqk|x[k2]|)pn=∑n=0∞(a2n∑k=02nqk|x[k2]|)p2n+∑n=0∞(a2n+1∑k=02n+1qk|x[k2]|)p2n+1≤∑n=0∞(a2n[q2n|xn|+∑k=02n(q2k+q2k+1)|xk|])pn+∑n=0∞(a2n+1(∑k=02n+1(q2k+q2k+1)|x[k2]|)pn≤2h−1(∑n=0∞(Can∑k=0nqk|xk|)pn+∑n=0∞(2anC∑k=0nqk|xk|)pn)+∑n=0∞(2Can∑k=0nqk|xk|)pn≤2h−1(2h+1)Ch∑n=0∞(an∑k=0nqk|xk|)pn+2hCh∑n=0∞(an∑k=0nqk|xk|)pn≤(22h−1+2h−1+2h)Ch∑n=0∞(an∑k=0nqk|xk|)pn<∞,

then ( x[n2] ) ∈ ces((a_n), (p_n), (q_n)).

By using Theorem (2.17), we can get the following corollary:

Corollary 3.2

Let conditions (a1), (a2) and (a3) be satisfied, then S_{ces((a_n),(p_n),(q_n))} is an operator ideal.

4 Topological properties

The following question arises naturally; for which sufficient conditions (not necessary) on the pre-modular (sss) E_ρ, the ideal of the finite rank operators on the class of Banach spaces is dense in S_{E_ρ}? This gives a negative answer of Rhoades [9] open problem about the linearity of E_ρ type spaces (S_{E_ρ}).

Theorem 4.1

ces((a_n), (p_n), (q_n)) is a pre-modular (sss), if conditions (a1), (a2) and (a3) are satisfied.

Proof

Define a functional ρ on ces((a_n), (p_n), (q_n)) by ρ(x) = ∑n=0∞ (a_n ∑k=0n q_k|x_k|)^p_n.

Clearly, ρ(x) ≥ 0 and ρ(x) = 0 ⇔ x = θ.
There is a number L = max{1, sup_n|λ|^p_n} ≥ 1 with ρ(λ x) ≤ L|λ|ρ(x) for all x ∈ ces((a_n), (p_n)) and λ ∈ ℝ.
We have the inequality ρ(x + y) ≤ 2^h−1(ρ(x) + ρ(y)) for all x, y ∈ ces((a_n), (p_n)).
It is clear from (2) theorem 3.1.
It is clear from (3) theorem 3.1, that K₀ ≥ (2^2h−1 + 2^h−1 + 2^h)C^h ≥ 1.
It is clear that F = ces((a_n), (p_n)).
There exists a constant 0 < ξ ≤ sup_n|λ|^p_n−1 such that ρ(λ, 0, 0, 0, …) ≥ ξ|λ|ρ(1, 0, 0, 0, …) for any λ ≠ 0 and ξ > 0, when λ = 0.

We state the following theorem without proof, this can be established using standard technique.

Theorem 4.2

The function g(P) = ϱ(si(P))i=0∞ is a pre-quasi norm on S_{E_ϱ}, where E_ϱ is a pre-modular (sss).

Lemma 4.3

If E_ρ is a pre-modular (sss) and (x_n) ∈ E_ρ is a monotonic decreasing sequence of positive reals, then

ρ(0,0,0,...,0⏞2n,xn,xn+1,xn+2,...)≤K0ρ(0,0,0,...,0⏞n,xn,xn+1,xn+2,...).

Proof

By using the Definition (2.15-iv, v) and since (x_n) ∈ E_ρ is monotonic decreasing we get

ρ(0,0,0,...,0⏞2n,xn,xn+1,xn+2,...)≤ρ(0,0,0,...,0⏞2n,xn,xn,xn+1,xn+1,...)≤K0ρ(0,0,0,...,0⏞n,xn,xn+1,xn+2,...).

Theorem 4.4

Let E_ρ be a pre-modular (sss). Then the linear space F(X, Y) is dense in S_{E_ρ}(X, Y), where g(T) = ρ (sn(T)n=0∞) .

Proof

It is easy to prove that every finite mapping T ∈ F(X, Y) belongs to S_{E_ρ}(X, Y), since e_m ∈ E_ρ for each m ∈ ℕ and E_ρ is a linear space then every finite mapping T ∈ F(X, Y) the sequence (sn(T))n=0∞ contains only finitely many numbers different from zero. To prove that S_{E_ρ}(X, Y) ⊆ F(X, Y), let T ∈ S_E(X, Y), then ρ( (sn(T))n=0∞ ) < ∞ and let ε ∈ (0, 1), at that point there exists a N ∈ ℕ-{0} such that ρ( (sn(T))n=N∞ ) < ε. While (s_n(T))_n∈ℕ is decreasing, we get

ρ((s2N(T))n=N+12N)≤ρ((sn(T))n=N+12N)≤ρ((sn(T))n=N∞)<ε. (2)

Hence, there exists A_N ∈ F(X, Y), rankA ≤ N and

ρ((∥T−A∥)n=2N+13N)≤ρ((∥T−A∥)n=N+12N)<ε. (3)

On considering

ρ((∥T−A∥)n=0N)<ε. (4)

We have to prove that ρ((sn(T−AN))n=0∞)→0 as N → ∞. By taking N = 8η, where η is a natural number. From Definition (2.15-iii) we have

d(T,AN)=ρ((sn(T−AN))n=0∞)=ρ[(s0(T−AN),s1(T−AN),...,s8η−1(T−AN),0,0,0,...)+(0,0,0,...,0⏞8η,s8η(T−AN),s8η+1(T−AN),...,s12η−1(T−AN),0,0,0,...)+(0,0,0,...,0⏞12η,s12η(T−AN),s12η+1(T−AN),...)]≤K2[ρ(s0(T−AN),s1(T−AN),...,s8η−1(T−AN),0,0,0,...)+ρ(0,0,0,...,0⏞8η,s8η(T−AN),s8η+1(T−AN),...,s12η−1(T−AN),0,0,0,...)+ρ(0,0,0,...,0⏞12η,s12η(T−AN),s12η+1(T−AN),...)]=K2[I1(N)+I2(η)+I3(η)].

Since s_n(A_N) = 0 for n ≥ N then

sn(T−AN)≤sn−N(T). (5)

By using Lemma 4.3, inequality (5) and Definition (2.15-iv), we get

I3(η)=ρ(0,0,0,...,0⏞12η,s12η(T−AN),s12η+1(T−AN),...)≤ρ(0,0,0,...,0⏞12η,s4η(T),s4η+1(T),...)≤K0ρ(0,0,0,...,0⏞6η,s4η(T),s4η+1(T),...)≤K02ρ(0,0,0,...,0⏞3η,s4η(T),s4η+1(T),...)≤K02ρ(0,0,0,...,0⏞3η,s3η(T),s3η+1(T),...)=K02ρ((sn(T))n=3η∞→0asη→∞.

Now using Lemma 4.3 and Definition (2.15-iv), we have

I2(η)=ρ(0,0,0,...,0⏞8η,s8η(T−AN),s8η+1(T−AN),...,s12η−1(T−AN),0,0,0,...)≤K0ρ(0,0,0,...,0⏞4η,s8η(T−AN),s8η+1(T−AN),...,s12η−1(T−AN),0,0,0,...)≤K0ρ(s0(T−AN),s1(T−AN),...,s8η−1(T−AN),0,0,0,...)=k0I1(N).

Finally we have to show that I₁(N) → 0 as N → ∞. By taking ε₁ = ε3LK for each n ≥ N₀(ε₁), using the inequalities (2), (3), (4) and Definition (2.15-ii and iii) then we have

I1(N)=ρ(s0(T−AN),s1(T−AN),...,sN−1(T−AN),0,0,0,...)≤K[ρ(s0(T−AN),s1(T−AN),...,sN0−1(T−AN),0,0,0,...)+ρ(0,0,0,...,0⏞N0,sN0(T−AN),sN0+1(T−AN),...,sN−1(T−AN),0,0,0,...)]≤K[ρ(∥T−AN∥,∥T−AN∥,...,∥T−AN∥,0,0,0,...)+ρ(0,0,0,...,0⏞N0,2sn(T),2sn(T),...,2sn(T),0,0,0,...)]≤K[ε1+2Lε1]<ϵ.

This completes the proof.

Corollary 4.5

F(X, Y) = S_{ces((a_n),(p_n),(q_n))}(X, Y), assuming that states (a1), (a2) and (a3) are fulfilled and the converse is not necessarily true.

Proof

It follows from Theorem (4.4) and ces((a_n), (p_n), (q_n)) is pre-modular (sss). For the converse part, since I₃ ∈ S_{ces((1),(1),(1))} but the condition (a2) is not satisfied which is a counter example. This establishes the proof.

From Corollary (4.5), we can say that if (a1), (a2) and (a3) are satisfied, then every compact operators would be approximated by finite rank operators and the converse is not necessarily true.

5 Completeness of the pre-quasi closed ideal components

For which sequence space ces((a_n), (p_n), (q_n)), the components of pre-quasi closed operator ideal S_{ces((a_n),(p_n),(q_n))} are complete?

Theorem 5.1

If X and Y are Banach spaces, (a1), (a2) and (a3) are satisfied, then (S_ces((a_n),(p_n),(q_n))_ρ, g), where ρ(x)=∑n=0∞(an∑k=0nqk|x(k)|)pn and g(T) = ρ( (sn(T))n=0∞ ) is a pre-quasi Banach operator ideal.

Proof

Since ces((a_n), (p_n), (q_n))_ρ is a pre-modular (sss), then the function g(T) = ρ( (sn(T))n=0∞ ) is a pre-quasi norm on S_{ces((a_n),(p_n),(q_n))_ρ}. Let (T_m) be a Cauchy sequence in S_{ces((a_n),(p_n),(q_n))_ρ}(X, Y), then by definition (2.15-vii) there exists a constant ξ > 0 and since L(X, Y)supseteq S_ces((a_n), (p_n), (q_n))_ρ}(X, Y), we get

g(Ti−Tj)=∑n=0∞(an∑k=0nqksk(Ti−Tj))pn≥∑n=0∞(anq0∥Ti−Tj∥)pn≥ξ∥Ti−Tj∥∑n=0∞(anq0)pn,

then (T_m)_m∈ℕ is a Cauchy sequence in L(X, Y). While the space L(X, Y) is a Banach space, so there exists T ∈ L(X, Y) with limm→∞ ∥T_m − T∥ = 0 and while (sn(Tm))n=0∞ ∈ ces((a_n), (p_n), (q_n))_ρ for each m ∈ ℕ, hence using definition (2.15) conditions (iii), (iv), (v) and ρ is continuous at θ, we get

g(T)=∑n=0∞(an∑k=0nqk(sk(T))pn=∑n=0∞(an∑k=0nqk(sk(T−Tm+Tm))pn≤K(∑n=0∞(an∑k=0nq[k2]s[k2](T−Tm)))+K(∑n=0∞(an∑k=0nq[k2]s[k2](Tm))pn)≤K∑n=0∞(an∑k=0nqk∥Tm−T∥)+KK0∑n=0∞(an∑k=0nqksk(Tm))pn<ε,

We have (sn(T))n=0∞ ∈ ces((a_n), (p_n), (q_n))_ρ, then T ∈ S_{ces((a_n),(p_n),(q_n))_ρ}(X, Y).

Theorem 5.2

If X and Y are normed spaces, (a1), (a2) and (a3) are satisfied, then (S_{ces((a_n),(p_n),(q_n))_ρ}, g), where ρ(x)=∑n=0∞(an∑k=0nqk|x(k)|)pn and g(T) = ρ( (sn(T))n=0∞ ) is a pre-quasi closed operator ideal.

Proof

Since ces((a_n), (p_n), (q_n))_ρ is a pre-modular (sss), then the function g(T) = ρ( (sn(T))n=0∞ ) is a pre-quasi norm on S_{ces((a_n),(p_n),(q_n))_ρ}. Let T_m ∈ S_{ces((a_n),(p_n),(q_n))_ρ}(X, Y) for all m ∈ ℕ and limm→∞ g(T_m − T) = 0, then by utilizing definition (2.15-vii) there exists a constant ξ > 0 and since L(X, Y)supseteq S_{ces((a_n),(p_n),(q_n))_ρ}(X, Y), we get

g(T−Tm)=∑n=0∞(an∑k=0nqksk(T−Tm))pn≥∑n=0∞(anq0∥T−Tm∥)pn≥ξ∥T−Tm∥∑n=0∞(anq0)pn,

then (T_m)_m∈ℕ is a convergent sequence in L(X, Y). While (sn(Tm))n=0∞ ∈ ces((a_n), (p_n), (q_n))_ρ for each m ∈ ℕ, hence using definition (2.15) conditions (iii), (iv), (v) and ρ is continuous at θ, we obtain

we have (sn(T))n=0∞ ∈ ces((a_n), (p_n), (q_n))_ρ, then T ∈ S_{ces((a_n),(p_n),(q_n))_ρ}(X, Y).

6 Minimum pre-quasi Banach operator ideal

We give here the sufficient conditions on the generalized Cesáro sequence space defined by weighted means such that the pre-quasi operator ideal formed by the sequence of approximation numbers and this sequence space is strictly contained for different weights and powers.

Theorem 6.1

For any infinite dimensional Banach spaces X, Y and for any 1 < 1<pn(1)<pn(2), 0<an(2)≤an(1) and 0<qn(2)≤qn(1) for all n ∈ ℕ, it is true that

Sces((an(1)),(pn(1)),(qn(1)))app(X,Y)⫋Sces((an(2)),(pn(2)),(qn(2)))app(X,Y)⫋L(X,Y).

Proof

Let X and Y be infinite dimensional Banach spaces and for any 1<pn(1)<pn(2), 0<an(2)≤an(1) and 0<qn(2)≤qn(1) for all n ∈ ℕ, if T ∈ Sces((an(1)),(pn(1)),(qn(1)))app (X, Y), then (αn(T))∈ces((an(1)),(pn(1)),(qn(1))). We have

∑n=0∞(an(2)∑k=0nqk(2)αk(T))pn(2)<∑n=0∞(an(1)∑k=0nqk(1)αk(T))pn(1)<∞,

hence T ∈ Sces((an(2)),(pn(2)),(qn(2)))app (X, Y). Next, if we take (αn(T))n=0∞ such that an(1)∑k=0nqk(1)αk(T)=1n+1pn(1), one can find T ∈ L(X, Y) with

∑n=0∞(an(1)∑k=0nqk(1)αk(T))pn(1)=∑n=0∞1n+1=∞,

and

∑n=0∞(an(2)∑k=0nqk(2)αk(T))pn(2)≤∑n=0∞(an(1)∑k=0nqk(1)αk(T))pn(2)=∑n=0∞(1n+1)pn(2)pn(1)<∞.

Hence T does not belong to Sces((an(1)),(pn(1)),(qn(1)))app (X, Y) and T ∈ Sces((an(2)),(pn(2)),(qn(2)))app (X, Y).

It is easy to verify that Sces((an(2)),(pn(2)),(qn(2)))app (X, Y) ⊂ L(X, Y). Next, if we take (αn(T))n=0∞ such that an(2)∑k=0nqk(2)αk(T)=1n+1pn(2). One can find T ∈ L(X, Y) such that T does not belong to Sces((an(2)),(pn(2)),(qn(2)))app (X, Y). This completes the proof.

Corollary 6.2

For any infinite dimensional Banach spaces X, Y and 1 < p < q < ∞, then

Scespapp(X,Y)⫋Scesqapp(X,Y)⫋L(X,Y).

In this part, we give the conditions for which the the pre-quasi Banach Operator ideal Sces((an),(pn),(qn)app is minimum.

Theorem 6.3

If conditions (a1), (a2), (a3) and (a_n ∑j=0n q_j) ∉ ℓ^(p_n) are satisfied, then the pre-quasi Banach Operator ideal Sces((an),(pn),(qn)app is minimum.

Proof

Let conditions (a1), (a2), (a3) and (a_n ∑j=0n q_j) ∉ ℓ^(p_n) be satisfied. Then ( Sces((an),(pn),(qn)app , g), where g(T)=[∑i=0∞(ai∑j=0iqjαj(T))pi]1h is a pre-quasi Banach Operator ideal. Let X and Y be any two Banach spaces. Suppose that Sces((an),(pn),(qn)app = L(X, Y), then there exists a constant C > 0 such that g(T) ≤ C∥T∥ for all T ∈ L(X, Y). Assume that X and Y be infinite dimensional Banach spaces. Hence by Dvoretzky’s theorem [20] for m ∈ ℕ, we have quotient spaces X/N_m and subspaces M_m of Y which can be mapped onto ℓ2m by isomorphisms H_m and A_m such that ∥Hm∥∥Hm−1∥≤2 and ∥Am∥∥Am−1∥≤2. Let I_m be the identity map on ℓ2m , Q_m be the quotient map from X onto X/N_m and J_m be the natural embedding map from M_m into Y. Let u_n be the Bernstein numbers [21] then

1=un(Im)=un(AmAm−1ImHmHm−1)≤∥Am∥un(Am−1ImHm)∥Hm−1∥=∥Am∥un(JmAm−1ImHm)∥Hm−1∥≤∥Am∥dn(JmAm−1ImHm)∥Hm−1∥=∥Am∥dn(JmAm−1ImHmQm)∥Hm−1∥≤∥Am∥αn(JmAm−1ImHmQm)∥Hm−1∥,

for 0 ≤ i ≤ m. Now

∑j=0iqj≤∑j=0i∥Am∥qjαj(JmAm−1ImHmQm)∥Hm−1∥⇒ai∑j=0iqj≤∥Am∥(ai∑j=0iqjαj(JmAm−1ImHmQm))∥Hm−1∥⇒(ai∑j=0iqj)pi≤(∥Am∥∥Hm−1∥)pi(ai∑j=0iqjαj(JmAm−1ImHmQm))pi.

Therefore,

[∑i=0m(ai∑j=0iqj)pi]1h≤L∥Am∥∥Hm−1∥[∑i=0m(ai∑j=0iqjαj(JmAm−1ImHmQm))pi]1h⇒[∑i=0m(ai∑j=0iqj)pi]1h≤L∥Am∥∥Hm−1∥[∑i=0m(ai∑j=0iqjαj(JmAm−1ImHmQm))pi]1h⇒[∑i=0m(ai∑j=0iqj)pi]1h≤L∥Am∥∥Hm−1∥g(JmAm−1ImHmQm)⇒[∑i=0m(ai∑j=0iqj)pi]1h≤LC∥Am∥∥Hm−1∥∥JmAm−1ImHmQm∥⇒[∑i=0m(ai∑j=0iqj)pi]1h≤LC∥Am∥∥Hm−1∥∥JmAm−1∥∥Im∥∥HmQm∥=LC∥Am∥∥Hm−1∥∥Am−1∥∥Im∥∥Hm∥⇒[∑i=0m(ai∑j=0iqj)pi]1h≤4LC,

for some L ≥ 1. We arrive at a contradiction, since m is an arbitrary and (a_n ∑j=0n q_j) ∉ ℓ^(p_n). Thus X and Y both cannot be infinite dimensional when Sces((an),(pn),(qn)app = L(X, Y). Hence, the result.

Theorem 6.4

If conditions (a1), (a2), (a3) and (a_n ∑j=0n q_j) ∉ ℓ^(p_n) are satisfied, then the pre-quasi Banach Operator ideal Sces((an),(pn),(qn)Kol is minimum.

Corollary 6.6

If 1 < p < ∞, then the quasi Banach Operator ideal Scespapp is minimum.

Corollary 6.6

If 1 < p < ∞, then the quasi Banach Operator ideal ScespKol is minimum.

7 Pre-quasi simple Banach operator ideal

The following question arises naturally; for which generalized Cesáro sequence space defined by weighted means, the pre-quasi Banach ideal is simple?

Theorem 7.1

For any infinite dimensional Banach spaces X, Y. If (pn(1)), (pn(2)) are bounded sequences with 1<pn(1)<pn(2), 0<an(2)≤an(1) and 0 < qn(2)≤qn(1) for all n ∈ ℕ, then

L(Sces((an(2)),(pn(2)),(qn(2))),Sces((an(1)),(pn(1)),(qn(1))))=Ψ(Sces((an(2)),(pn(2)),(qn(2))),Sces((an(1)),(pn(1)),(qn(1)))).

Proof

Suppose that there exists T ∈ L(Sces((an(2)),(pn(2)),(qn(2))),Sces((an(1)),(pn(1)),(qn(1)))) which is not approximable. According to Lemma (2.6), we can find X ∈ L(Sces((an(2)),(pn(2)),(qn(2))),Sces((an(2)),(pn(2)),(qn(2)))) and B ∈ L(Sces((an(1)),(pn(1)),(qn(1))),Sces((an(1)),(pn(1)),(qn(1)))) with BTX I_k = I_k. Then it follows for all k ∈ ℕ that

∥Ik∥Sces((an(1)),(pn(1)),(qn(1)))=(∑n=0∞(an(1)∑i=0nqi(1)si(Ik))pn(1))1suppn(1)≤∥BTX∥∥Ik∥Sces((an(2)),(pn(2)),(qn(2)))≤(∑n=0∞(an(2)∑i=0nqi(2)si(Ik))pn(2))1suppn(2).

This contradicts Theorem(6.1). Hence T ∈ Ψ (Sces((an(2)),(pn(2)),(qn(2))),Sces((an(1)),(pn(1)),(qn(1)))) , which finishes the proof.

Corollary 7.2

For any infinite dimensional Banach spaces X, Y. If (pn(1)), (pn(2)) are bounded sequences with 1<pn(1)<pn(2), 0<an(2)≤an(1) and 0 < qn(2)≤qn(1) for all n ∈ ℕ, then

L(Sces((an(2)),(pn(2)),(qn(2))),Sces((an(1)),(pn(1)),(qn(1))))=LC(Sces((an(2)),(pn(2)),(qn(2))),Sces((an(1)),(pn(1)),(qn(1)))).

Proof

Since every approximable operator is compact.

Theorem 7.3

For a bounded sequence (p_n) with 1 < p_n < ∞, q_n > 0 and a_n > 0 for all n ∈ ℕ, the pre-quasi Banach space S_{ces((a_n),(p_n),(q_n))} is simple.

Proof

Suppose that the closed ideal L_C(S_{ces((a_n),(p_n),(q_n))}) contains an operator T which is not approximable. According to Lemma (2.6), we can find X, B ∈ L(S_{ces((a_n),(p_n),(q_n))}) with BTX I_k = I_k. This means that I_{S_{ces((a_n),(p_n),(q_n))}} ∈ L_C(S_{ces((a_n),(p_n),(q_n))}). Consequently L(S_{ces((a_n),(p_n),(q_n))}) = L_C(S_{ces((a_n),(p_n),(q_n))}). Therefore Ψ(S_{ces((a_n),(p_n),(q_n))}) is the only non-trivial closed ideal in L(S_{ces((a_n),(p_n),(q_n))}).

8 Eigenvalues of s-type operators

We give here the sufficient conditions on the generalized Cesáro sequence space defined by weighted means such that the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space is strictly contained in the class of all bounded linear operators whose sequence of eigenvalues belongs to this sequence space.

Notation 8.1

SEλ:={SEλ(X,Y);XandYare Banach Spaces},whereSEλ(X,Y):={T∈L(X,Y):((λi(T))n=0∞∈Eand∥T−λn(T)∥is not invertible for alln∈N}.

Theorem 8.2

For any infinite dimensional Banach spaces X and Y. If conditions (a1), (a2), and (a3) are satisfied with inf_n q_n > 0 and inf_n(n + 1)a_n > 0, then

Sces((an),(pn),(qn))(X,Y)⫋Sces((an),(pn),(qn))λ(X,Y).

Proof

Let T ∈ S_{ces((a_n),(p_n),(q_n))}(X, Y), then (sn(T))n=0∞ ∈ ces((a_n), (p_n), (q_n)). Since inf_n q_n > 0, inf_n(n + 1)a_n > 0 and (p_n) is increasing bounded sequence with p_n > 1 for all n ∈ ℕ, we have

∑n=0∞(an∑k=0nqksk(T))pn≥infn(infn[(n+1)anqn])pn∑n=0∞[sn(T)]pn.

Hence (sn(T))n=0∞ ∈ ℓ^(p_n), so lim_n→∞ s_n(T) = 0. Suppose ∥T − s_n(T)∥ is invertible for all n ∈ ℕ, then ∥T − s_n(T)∥⁻¹ exists and bounded for all n ∈ ℕ. This gives lim_n→∞∥T − s_n(T)∥⁻¹ = ∥T∥⁻¹ exists and bounded. Since (S_{ces((a_n),(p_n),(q_n))}, g) is a pre-quasi operator ideal, we have

I=TT−1∈Sces((an),(pn),(qn))(X,Y)⇒(sn(I))n=0∞∈ces((an),(pn),(qn))⇒limn→∞sn(I)=0.

But lim_n→∞ s_n(I) = 1. This is a contradiction, then ∥T − s_n(T)∥ is not invertible for all n ∈ ℕ. Therefore the sequence (sn(T))n=0∞ is the eigenvalues of T. Next, on considering (sn(T))n=0∞ such that a_n ∑k=0n q_k s_k(T) = 1n+1pn , we find T ∈ L(X, Y) with

∑n=0∞(an∑k=0nqksk(T))pn=∑n=0∞1n+1=∞,

and if we take (λn(T))n=0∞ such that an∑k=0nqkλk(T)=1n+1. Hence T does not belong to S_{ces((a_n),(p_n),(q_n))}(X, Y) and T ∈ Sces((an),(pn),(qn))λ(X,Y) (X, Y). This finishes the proof.

awad_bakry@hotmail.com

Acknowledgements

The authors thank the anonymous referees for their constructive suggestions and helpful comments which led to significant improvement of the original manuscript of this paper.

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Received: 2019-04-20

Accepted: 2019-10-19

Published Online: 2019-12-31

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0135

Keywords for this article

pre-quasi operator ideal; s-numbers; generalized Cesáro sequence space; weighted means; simple Banach space

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