  Answering a question of G. Fici, we give an $S$-adic characterization of
thefamily of infinite LSP words, that is, the family of infinite words having
all their left special factors as prefixes.More precisely we provide a finite
set of morphisms $S$ and an automaton ${\cal A}$ such that an infinite word is
LSP if and only if it is $S$-adic and one of its directive words is
recognizable by ${\cal A}$.Then we characterize the endomorphisms that preserve
the property of being LSP for infinite words.This allows us to prove that there
exists no set $S'$ of endomorphisms for which the set of infinite LSP words
corresponds to the set of $S'$-adic words. This implies that an automaton is
required no matter which set of morphisms is used.
