Small operator ideals formed by s numbers on generalized Cesáro and Orlicz sequence spaces

Area of Science:

• Functional Analysis
• Operator Theory
• Sequence Space Theory

Background:

• Operator ideals are fundamental in functional analysis, characterizing classes of linear operators.
• Generalized Cesáro and Orlicz sequence spaces are significant in studying operator properties.
• Understanding the structure of operator ideals is crucial for classifying operators.

Purpose of the Study:

• To establish conditions for generalized Cesáro and Orlicz sequence spaces to generate operator ideals.
• To investigate the properties of these generated operator ideals, including completeness and density.
• To explore inclusion relations between these operator ideals and their duals.

Main Methods:

• Utilizing the theory of generalized Cesáro and Orlicz sequence spaces.
• Applying concepts from operator ideal theory, including s-numbers and approximation numbers.
• Proving properties such as completeness and density within the framework of pre-quasi Banach operator ideals.

Main Results:

• Sufficient conditions are established for the class of bounded linear operators to form an operator ideal.
• The generated operator ideal is shown to be a pre-quasi Banach operator ideal with finite-dimensional operators as a dense subset.
• Inclusion relations for the operator ideals and their duals are derived.
• The operator ideal formed by approximation numbers is demonstrated to be small under specific conditions.

Conclusions:

• The research successfully constructs and characterizes novel operator ideals based on generalized Cesáro and Orlicz sequence spaces.
• The findings contribute to the deeper understanding of operator ideal theory and its relationship with sequence spaces.
• The study provides a foundation for further investigations into the properties and applications of these operator ideals.