Fuzzy Operator Ideals

Abstract

Although the systematic emergence of the fuzzy functional analysis theory has started in the last few years, we are starting to construct a new theory of fuzzy operator ideals inspired by the classical (crisp) operator ideals theory. We define the concept of fuzzy operator ideal and present some basic examples and introduce a significant class of fuzzy operator ideals, namely the class of absolutely fuzzy p-summing between arbitrary complete fuzzy normed spaces, which is a natural generalization of the notion of absolutely (crisp) p-summing between Banach spaces defined by Albrecht Pietsch. We initiate to define fuzzy norm of the aforementioned notion and prove its fuzzy norm is fuzzy real number. It shows that the resulting class of fuzzy bounded operators is a complete fuzzy normed fuzzy operator ideal. It then establishes an important fundamental characterization of bsolutely fuzzy p-summing. This is done by proving a fuzzy version of Pietsch Domination Theorems. Finally, the paper presents some open problems which the researchers finds interesting.

Keywords

• operator ideals
• fuzzy functional analysis
• fuzzy real analysis
• pietsch domination theorem