Fuzzy bounded operators with application to Radon transform

Abstract This paper is focused on developing the means to extend the range of application of the inverse Radon transform by enlarging the domain of definition of the Radon operator, namely from a specific Banach space to a more general fuzzy normed linear space. This is done by studying different types of fuzzy bounded linear operators acting between fuzzy normed linear spaces. The motivation for considering this type of spaces comes from the existence of an equivalence between the probabilistic metric spaces and fuzzy metric spaces, in particular fuzzy normed linear spaces. We mention that many notions and results belonging to classical metric spaces could also be found in this general context. Moreover, this setup allows to develop applications as diverse as: image processing, data compression, signal processing, computer graphics, etc. The class of operators that best fits the intended purpose is the class of strongly fuzzy bounded linear operators. The main results about this family of operators use the fact that the space of such operators becomes a normed algebra. An extension of the classical norm of a bounded linear operator between two normed spaces to the norm of strongly fuzzy bounded linear operators acting between fuzzy normed linear spaces is proved. A version of the classical Banach-Steinhaus theorem for strongly fuzzy bounded linear operators is given. A sufficient condition for the limit of a sequence of strongly fuzzy bounded linear operators to be strongly fuzzy bounded is shown. The adjoint operator of a strongly fuzzy bounded linear operator is a classic bounded linear operator. The class of neighborhood fuzzy bounded linear operators are studied as well, being established connections with two other classes of operator, namely the class of fuzzy bounded linear operators and strongly fuzzy bounded linear operators.