الفهرس | Only 14 pages are availabe for public view |
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 denitions 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 denitions 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 dened 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 dened 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 dened by a sequence of modulus functions. Dierent 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 dierence sequence spaces dened 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]. |