Common hypercyclic vectors for unilateral weighted shifts on ℓ2

Each w∈ℓ∞ defines a bacwards weighted shift Bw:ℓ2→ℓ2. A vector x∈ℓ2 is \textit{hypercyclic} for Bw if the set of forward iterates of x is dense in ℓ2. For each such w, the set HC(w) consisting of all vectors hypercyclic for Bw is Gδ. The set of \textit{common hypercyclic vectors} for a set W⊆ℓ∞ is the set HC∗(W)=⋂w∈WHC(w). We show that HC∗(W) can be made arbitrarily complicated by making W sufficiently complex, and that even for a Gδ set W the set HC∗(W) can be non-Borel. Finally, by assuming the continuum hypothesis or Martin's axiom, we are able to construct a set W such that HC∗(W) does not have the property of Baire.