The semigroup of weighted composition operators $(W_n)_{ n\in \mathbb{ N}}$, defined by $$W_nf(z)=(1+z+\cdots +z^n)f(z^n),$$ acts on the classical Hardy-Hilbert space $H^{ 2}(\mathbb{ D})$, and exhibits intriguing connections with both the Riemann Hypothesis (RH) and the Invariant Subspace Problem (ISP). In this paper, we prove that the adjoint operators $W^{ \ast}_{ n}$, for $n\geq 2$, are Devaney chaotic, frequently hypercyclic and mixing. In particular, these operators are hypercyclic and discuss connections with the RH and invariant subspaces.
Categories: math.FA