Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces

In this paper, we give the sufficient conditions on Orlicz-Cesáro mean sequence spaces cesφ, where φ is an Orlicz function such that the class Scesφ of all bounded linear operators between arbitrary Banach spaces with its sequence of s−numbers which belong to cesφ forms an operator ideal. The completeness and denseness of its ideal components are specified and Scesφ constructs a pre-quasi Banach operator ideal. Some inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are explained. Moreover, we have presented the sufficient conditions on cesφ such that the pre-quasi Banach operator ideal generated by approximation number is small. The above results coincide with that known for cesp (1 < p < ∞).


Introduction
Throughout the paper, by , we mean the space of all real sequences, R the real numbers, and N = {0, 1, 2, . ..} and L(, ) the space of all bounded linear operators from a normed space  into a normed space .The operator ideals theory takes an importance in functional analysis, since it has numerous applications in fixed point theorem, geometry of Banach spaces, spectral theory, eigenvalue distributions theorem, etc.Some of the operator ideals in the class of normed spaces or Banach spaces in functional analysis are characterized by various scalar sequence spaces.For example the ideal of compact operators is defined by kolmogorov numbers and the space  0 of convergent to zero sequences.Pietsch [1] inspected the operator ideals framed by the approximation numbers and the classical sequence space ℓ  (0 <  < ∞).He proved that the ideals of Hilbert Schmidt operators and nuclear operators between Hilbert spaces are defined by ℓ 2 and ℓ 1 , respectively, and the sequence of approximation numbers.In [2], Faried and Bakery examined the operator ideals developed by generalized Cesáro, Orlicz sequence spaces ℓ  , and the approximation numbers.In [3], Faried and Bakery studied the operator ideals constructed by − numbers, generalized Cesáro and Orlicz sequence spaces ℓ  and show that the operator ideal formed by the previous sequence spaces and approximation numbers is small under certain conditions.Also summation process and sequences spaces applications are closely related to Korovkin type approximation theorems and linear positive operators studied by Costarelli and Vinti [4] and Altomare [5].The idea of this paper is to examine a generalized class    by using Orlicz-Cesáro mean sequence spaces   and the sequence of -numbers, for which    constructs an operator ideal.The components of    as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved.The inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are determined.Finally, we show that the prequasi Banach operator ideal formed by the approximation numbers and   is small under certain conditions.These results coincide with that known for   , (1 <  < ∞) in [3].Furthermore we give some examples which support our main results.

Definitions and Preliminaries
Definition 1 (see [6]).The sequence (  ()) ∞ =0 , for all  ∈ L(, ) is named an -function and the number   () is called the  ℎ number of  if the following are satisfied: (e) rank property: if rank() ≤  then   () = 0 for every  ∈ L(, ); (f) property of norming: where   is the identity operator on R  .
There are a few instances of -numbers; we notice the accompanying conditions: (1) The n-th approximation number, denoted by   (), is defined by   () = inf{ ‖ − ‖ :  ∈ L(, ) and rank() ≤ }. ( (3) The n-th Weyl number, denoted by   (), is defined by (4) The n-th Kolmogorov number, denoted by   (), is defined by (5) The n-th Gel'fand number, denoted by   (), is defined by   () =   (  ), where   is a metric injection from the space  to a higher space  ∞ (Ψ) for an adequate index set Ψ.This number is independent of the choice of the higher space  ∞ (Ψ).
(6) The n-th Chang number, denoted by   (), is defined by Remark 2 (see [6]).Among all the -number sequences characterized above, it is easy to check that the approximation number,   (), is the largest and the Hilbert number, ℎ  (), is the smallest -number sequence, i.e., ℎ  () ≤   () ≤   () for any bounded linear operator .If  is defined on a Hilbert space and compact, then all the -numbers correspond with the eigenvalues of ||, where || = ( * ) where   is the dual of .
Presently we express some known results of dual of an number sequence.
Definition 16 (see [10,11]).The operator ideal U fl {U(, );     ℎ } is a subclass of linear bounded operators such that its components U(, ) which are subsets of L(, ) fulfill the accompanying conditions: (i)   ∈ U where  indicates one dimensional Banach space, where U ⊂ L.
Given an Orlicz function , the Orlicz-Cesáro mean sequence spaces is defined by (  , ‖.‖) is a Banach space with the Luxemburg norm given by It seems that Orlicz-Cesáro mean sequence spaces   appeared for the first time in 1988, when Lim and Yee found their dual spaces [26].Recently Cui, Hudzik, Petrot, Suantai, and Szymaszkiewicz obtained important properties of spaces   [27].In 2007 Maligranda, Petrot, and Suantai showed that   is not B-convex, if  ∈ Δ 2 and   ̸ = 0 [28].The extreme points and strong -points of   have been characterized by Foralewski, Hudzik, and Szymaszkiewicz in [29].In the case when () =   , 1 ≤  < ∞, the space   is just a Cesáro sequence space   , with the norm given by It is well known that  1 = {0} [30].
We denote (E  , ) for the linear space E equipped with the metrizable topology generated by .
Theorem 25 (see [32]).If ,  are infinite dimensional Banach spaces and   is a monotonic decreasing sequence to zero, then there exists a bounded linear operator  such that Notations 26 (see [3]).
The concept of pre-quasi operator ideal which is more general than the usual classes of operator ideal.
Theorem 29 (see [3]).Every quasi norm on the ideal Ω is a pre-quasi norm on the ideal Ω.

Main Results
We give here the conditions on Orlicz-Cesáro mean sequence spaces   such that the class    of all bounded linear operators between arbitrary Banach spaces with its sequence of −numbers which belong to   forms an operator ideal.
Theorem 30.If  is an Orlicz function satisfying Δ 2 -condition and   > 1, then    is an operator ideal.
(1-ii) Let  ∈ R and  ∈   , and since  is convex and satisfying Δ 2 -condition, we get for some  > 0 that then  ∈   ; from (1-i) and (1-ii)   is a linear space.Since   ∈ ℓ  , for all  ∈ N and   > 1, then from Theorem 20, we get   ∈   , for all  ∈ N.
( We give the conditions on Orlicz-Cesáro mean sequence spaces   such that the ideal of the finite rank operators is dense in    (, ).Theorem 32.   (, ) = (X, ), if  is an Orlicz function satisfying Δ 2 -condition and   > 1.
We express the accompanying theorem without verification; these can be set up utilizing standard procedure.We give the sufficient conditions on Orlicz-Cesáro mean sequence spaces   such that the components of the prequasi operator ideal    are complete.
We now study some properties of the pre-quasi Banach operator ideal    .
Theorem 39.The pre-quasi Banach operator ideal (   , ) is injective, if the -number sequence is injective.
Theorem 41.The pre-quasi Banach operator ideal (   , ) is surjective, if the -number sequence is surjective.
Likewise, we have the accompanying inclusion relations between the pre-quasi Banach operator ideals.
Theorem 43. ( Hence the result is as follows.
We presently express the dual of the pre-quasi operator ideal formed by different − number sequences.
) is a prequasi Banach operator ideal.Let  and  be any two Banach spaces.Assume that     (, ) = L(, ), then there exists a constant  > 0 such that () ≤ ‖‖ for all  ∈ L(, ).Suppose that  and  are infinite dimensional Banach spaces.Then by Dvoretzky's theorem [8] for  ∈ N, we have quotient spaces /  and subspaces   of  which can be mapped onto ℓ  2 by isomorphisms   and   such that ‖  ‖‖ −1  ‖ ≤ 2 and ‖  ‖‖ −1  ‖ ≤ 2. Consider   be the identity map on ℓ  2 ,   be the quotient map from  onto /  , and   be the natural embedding map from   into .Let V  be the Bernstein numbers [7], then

Examples
We give some examples which support our main results.
Example 1.Let  be an Orlicz function; the subspace  ℎ  of all order continuous elements of   is defined as [ If  is an Orlicz function satisfying Δ 2 -condition and   > 1, then the following conditions are satisfied: (1)   ℎ  is an operator ideal.