The specification property and infinite entropy for certain classes of linear operators

We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces. It is known, from the work of Bartoll, Martínez-Giménez, Murillo-Arcila, and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We show that these forms of chaos (unsurprisingly) imply infinite topological entropy, but that (surprisingly) the converse does not hold.