Fuzzy compact operators and topological degree theory

Abstract

In this paper we first prove that the definition of fuzzy norm continuity and fuzzy boundedness of linear operators are equivalent. We show that the continuity assumption in the definition of compact linear operator is not necessary. We also point out that there is a gap in the proof of a theorem of Xiao and Zhu and we give a corrected version of the theorem such that all results based on the revised theorem remain true. Furthermore, we define a fuzzy product norm on the Cartesian product of two fuzzy normed spaces and prove a multiplicative property for the Leray–Schauder topological degree.