IDEALS OF SEQUENCES AND OPERATORS

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 Ces aro 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 su cient 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.