الفهرس 
PreliminariesOnly 14 pages are availabe for public view
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Abstract
The aim of this thesis is to study some operator ideals according
to that their sequences of s-numbers belong to certain sequence ideal.
For example if the sequence of approximation numbers of an operator
converges to zero then it belongs to the ideal of compact operators and
if this sequence is absolutely p-summing, then it belongs to the ideal
of Shatten Von Neumann operators.
The thesis consists of ve chapters:
Chapter 1
This chapter is an introductory chapter. It contains de nitions
and basic concepts that are used throughout this thesis. It is regarded
as a short survey of the basic needed material.
Chapter 2
Our goal in this chapter is to discuss the two concepts of operator
ideal and sequence ideal. The purpose of it is to present a short
survey of some needed de nitions and basic concepts of these two important
vital topics: operator ideal and sequence ideal.
Chapter 3
The aim of this chapter is studying the ideal of bounded linear
4
SUMMARY
operators between arbitrary Banach spaces whose approximation numbers
sequence belongs to the sequence space de ned by a sequence of
modulus functions. As a special case of our results, we form an operator
ideal using some well-known spaces like Cesaro sequence space and
Orlicz sequence space. In addition, we prove that the nite rank operators
are dense in the operator ideal formed by those spaces. Finally,
we show that the components of the operator ideal de ned by them
are complete. Our results generalize those in [27] by Faried and Bakery.
Some results of this chapter are:
Under submission [31].
Chapter 4
In this chapter, we introduce new generalized fractional order difference
sequence spaces which are de ned by a sequence of modulus
functions. Di erent algebraic and topological properties of these spaces
like linearity, completeness and solidity, etc are studied. Furthermore,
we drive necessary and sucient conditions for the inclusion relations
involving these spaces. Also, we prove that the ideal of bounded linear
operators between arbitrary Banach spaces whose approximation
numbers sequence belongs to those spaces can’t be obtained anyway
because they are not solid.
Some results of this chapter are published in:
Mathematical Sciences Letters, V. 6 N. 2, 2017 [29].
Chapter 5
This chapter is devoted to examine some general properties of the
generalized fractional order di erence sequence spaces de ned by a
sequence of Orlicz functions. In addition, we give some inclusion theorems
of these spaces. Furthermore, we show that the ideal of bounded
linear operators between arbitrary Banach spaces whose approximation
5
SUMMARY
numbers sequence belongs to those spaces can’t be obtained anyway
since they are not solid.
Some results of this chapter are accepted in:
International Journal of Advancement in Engineering Technology, Man-
agement and Applied Science [30].