Fuzzy Soft Prenormal Operators

Abstract

This paper introduces and systematically investigates a novel class of operators called fuzzy soft prenormal operators (FSP) within the framework of fuzzy soft Hilbert spaces. Motivated by the need to extend classical operator theory to handle uncertainty and imprecision, we develop this class as a meaningful generalization of fuzzy soft normal operators. We establish several fundamental properties and characterizations, demonstrating that FSP operators preserve essential spectral features while offering enhanced flexibility in modeling operator behavior under fuzzy and parametric uncertainty. Key results include: the closure properties of FSP operators under addition and multiplication under specific commutation conditions; the invariance of the FSP property under translation by scalar multiples of the identity; topological closure in the strong operator topology; and the significant theorem that every fuzzy soft isometry satisfying the FSP condition is necessarily unitary. Furthermore, we explore the relationships between fuzzy soft prenormal operators and other established classes such as fuzzy soft normal, self-adjoint, and unitary operators. The theoretical contributions presented here not only enrich the landscape of fuzzy soft operator theory but also provide a robust foundation for potential applications in mathematical physics, engineering, and decision-making under uncertainty.