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

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}, $$\begin{array}{} \displaystyle S_{E}(X, Y):=\Big\{T\in L(X, Y):((s_{n}(T))_{n=0}^{\infty}\in E\Big\}, \end{array}$$ of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, 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.


Introduction
Through the paper L(X, Y) = T : X → Y; T is a bounded linear operator ; Xand Y are 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 aftere ect of the enormous applications in geometry of Banach spaces, spectral theory, theory of eigenvalue distributions and xed 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 de ned by di erent scalar sequence spaces. For example the ideal of compact operators is de ned by the space c of null sequence and Kolmogorov numbers. Pietsch [1], examined the quasi-ideals formed by the approximation numbers and classical sequence space p ( < p < ∞). He proved that the ideals of nuclear operators and of Hilbert Schmidt operators between Hilbert spaces are de ned by and respectively. He proved that the class of all nite rank operators are dense in the Banach quasi-ideal and the algebra L( p ), where ( ≤ 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 di erent powers i.e., for any in nite dimensional Banach spaces X, Y and for any q > p > , it is true that S app p (X, Y) S app q (X, Y) L(X, Y). In [4], Faried and Bakery studied the operator ideals constructed by approximation numbers, generalized Cesáro and Orlicz sequence spaces 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 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 di erent 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 de ned by weighted means), we give su cient (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 nite 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 di erent 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 su cient 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.

De nitions and preliminaries
De nition 2.1. [10] An s-number function is a map de ned on L(X, Y) which associate to each operator T ∈ L(X, Y) a non-negative scaler sequence (sn(T)) ∞ n= assuming that the taking after states are veri ed: (1) For all T ∈ Ω(X, Y), g(T) ≥ and g(T) = if and only if T = , (2) there exists a constant M ≥ such that g(λT) ≤ M|λ|g(T), for all T ∈ Ω(X, Y) and λ ∈ R, (3) there exists a constant K ≥ such that g(T + T ) ≤ K[g(T ) + g(T )], for all T , T ∈ Ω(X, Y), (4) there exists a constant C ≥ such that if T ∈ L(X , X), P ∈ Ω(X, Y) and R ∈ L(Y , Y ) then g(RPT) ≤ C R g(P) T , where X and Y are normed spaces.
Theorem 2.12. [5] Every quasi norm on the ideal Ω is a pre-quasi norm on the ideal Ω.

De nition 2.15. [5] A subclass of the (sss) called a pre-modular (sss) assuming that we have a map ρ
for each x ∈ E and scalar λ, we get a real number L ≥ for which ρ(λx) ≤ |λ|Lρ(x), 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 Eρ. Moreover condition (1) in de nition (2.14) and condition (vi) in de nition (2.15) explain that (en) n∈N is a Schauder basis of Eρ.

Theorem 2.17. [5] If E is a (sss), then S E is an operator ideal.
Throughout, we denote en = { , , ..., , , , ...} where 1 appears at the n th place for all n ∈ N and the given inequality will be used in the sequel: where H = max{ , h− }, h = sup n pn and pn ≥ for all n ∈ N. See [13].

Linear problem
We examine here the operator ideals created by s−numbers also generalized Cesáro sequence space de ned by weighted means such that those classes of all bounded linear operators T between arbitrary Banach spaces with (sn(T)) n∈N in this sequence space type an ideal operator. (1-ii) let λ ∈ R, x ∈ ces((an), (pn), (qn)) and since (pn) is bounded, we have Hence em ∈ ces((an), (pn), (qn)).
By using Theorem (2.17), we can get the following corollary:

Topological properties
The following question arises naturally; for which su cient conditions (not necessary) on the pre-modular (sss) Eρ, the ideal of the nite rank operators on the class of Banach spaces is dense in S Eρ ? This gives a negative answer of Rhoades [9] open problem about the linearity of Eρ type spaces (S Eρ ). (ii) There is a number L = max , sup n |λ| pn ≥ with ρ(λx) ≤ L|λ|ρ(x) for all x ∈ ces((an), (pn)) and λ ∈ R.
(iv) It is clear from (2)  We state the following theorem without proof, this can be established using standard technique.

Theorem 4.4. Let Eρ be a pre-modular (sss). Then the linear space F(X, Y) is dense in S Eρ (X, Y), where g(T) = ρ(sn(T) ∞ n= ).
Proof: It is easy to prove that every nite mapping T ∈ F(X, Y) belongs to S Eρ (X, Y), since em ∈ Eρ for each m ∈ N and Eρ is a linear space then every nite mapping T ∈ F(X, Y) the sequence (sn(T)) ∞ n= contains only nitely many numbers di erent from zero. To prove that S Eρ (X, Y) ⊆ F(X, Y), let T ∈ S E (X, Y), then ρ((sn(T)) ∞ n= ) < ∞ and let ε ∈ ( , ), at that point there exists a N ∈ N − { } such that ρ((sn(T)) ∞ n=N ) < ε. While (sn(T)) n∈N is decreasing, we get Hence, there exists A N ∈ F(X, Y), rankA ≤ N and On considering We have to prove that ρ((sn(T − A N )) ∞ n= ) → as N → ∞. By taking N = η, where η is a natural number. From De nition (2.15-iii) we have  Proof: It follows from Theorem (4.4) and ces((an), (pn), (qn)) is pre-modular (sss). For the converse part, since I ∈ S ces(( ),( ),( )) but the condition (a2) is not satis ed which is a counter example. This establishes the proof. From Corollary (4.5), we can say that if (a1), (a2) and (a3) are satis ed, then every compact operators would be approximated by nite rank operators and the converse is not necessarily true.

Minimum pre-quasi Banach operator ideal
We give here the su cient conditions on the generalized Cesáro sequence space de ned 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 di erent weights and powers.
Theorem 6.1. For any in nite dimensional Banach spaces X, Y and for any < p ( ) n < p ( ) n , < a ( ) n ≤ a ( ) n and < q ( ) n ≤ q ( ) n for all n ∈ N, it is true that Proof: Let X and Y be in nite dimensional Banach spaces and for any < p ( ) Y). Next, if we take (αn(T)) ∞ n= such that a ( ) Hence T does not belong to S app ces((a ( ) n ),(p ( ) n ),(q ( ) n )) (X, Y) and T ∈ S app ces((a ( ) n ),(p ( ) n ),(q ( ) n )) (X, Y).