  We prove that, under certain conditions, uniform weak mixing (to zero) of the
bounded sequences in Banach space implies uniform weak mixing of its tensor
product. Moreover, we prove that ergodicity of tensor product of the sequences
in Banach space implies its weak mixing. Applications of the obtained results,
we prove that tensor product of uniquely $E$-weak mixing $C^*$-dynamical
systems is also uniquely $E$-weak mixing as well.
