Spaces that admit hypercyclic operators with hypercyclic adjoints

A continuous linear operator $T:X\to X$ is hypercyclic if there is an $x\in X$ such that the orbit $\{ T^n x\}_{n\geq 0}$ is dense. A result of H. Salas shows that any infinite-dimensional separable Hilbert space admits a hypercyclic operator whose adjoint is also hypercyclic. It is a natural question to ask for what other spaces $X$ does $\mathcal{L}(X)$ contain such an operator. We prove that for any infinite-dimensional Banach space $X$ with a shrinking symmetric basis, such as $c_0$ and any $\ell_p$ $(1<p<\infty)$, there is an operator $T:X \to X$, where both $T$ and $T':X'\to X'$ are hypercyclic.