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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 newsufficient condition for cyclicity of a weighted bilateral shift is proved,which implies, in particular, that any compact weighted bilateral shift iscyclic.

A weighted bilateral shift with cyclic square is supercyclic