A weighted bilateral shift with cyclic square is supercyclic

It is shown that for a bounded weighted bilateral shift $T$ acting on ${\mathrm{\ell }}_p(\mathbb{Z})$ for $1\le p\le 2$ supercyclicity of $T$, weak supercyclicity of $T$, cyclicity of $T\oplus T$ and cyclicity of $T^2$ are equivalent. A new sufficient condition for cyclicity of a weighted bilateral shift is proved, which implies, in particular, that any compact weighted bilateral shift is cyclic.