  We prove that reiteratively hypercyclic operators have perfect spectrum.
Consequently, it follows that there exist separable infinite dimensional Banach
spaces that do not support any reiteratively hypercyclic operator. For this, we
study $\mathcal F$-recurrence and almost $\mathcal {F}$-recurrence of operators
for general families and in particular for a special class of families, called
block families.
