  It is shown that for a bounded weighted bilateral shift $T$ acting on
$\ell_p(\Z)$ for $1\leq p\leq 2$ 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.
