  In this paper we introduce and study semigroups of operators on spaces of
fuzzy-number-valued functions, and various applications to fuzzy differential
equations are presented. Starting from the space of fuzzy numbers, many new
spaces sharing the same properties are introduced. We derive basic operator
theory results on these spaces and new results in the theory of semigroups of
linear operators on fuzzy-number kind spaces. The theory we develop is used to
solve classical fuzzy systems of differential equations, including, for
example, the fuzzy Cauchy problem and the fuzzy wave equation. These tools
allow us to obtain explicit solutions to fuzzy initial value problems which
bear explicit formulas similar to the crisp case, with some additional fuzzy
terms which in the crisp case disappear. The semigroup method displays a clear
advantage over other methods available in the literature (i.e., the level set
method, the differential inclusions method and other "fuzzification" methods of
the real-valued solution) in the sense that the solutions can be easily
constructed, and that the method can be applied to a larger class of fuzzy
differential equations that can be transformed into an abstract Cauchy problem.
