On the norm-closure of the class of hypercyclic operators

Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if ${\displaystyle f\left({\sigma }_W\left(T\right)\right)\cup \{z\in ℂ:\left|z\right|=1\}}$ is connected, where ${\displaystyle {\sigma }_W\left(T\right)}$ denotes the Weyl spectrum of T.