Apparently, the operators
$u(e_n) = e_{n+1}$
and
$u^\ast (e_n ) = e_{n-1}$
are called "unilateral shift oeprator".
Since they have to be called that (instead of calling them "the shift operator") I am wondering what the other types of shift operators are. I am struggling to come up with a definition for a multilateral shift operator.
What other shift operators are there?
The unitary shift operator on $l^{2}(\mathbb{Z})$ is not unilateral. This operator is the same as multiplication by $e^{i\theta}$ on $L^{2}[0,2\pi]$ with normalized inner product $$ (f,g) = \frac{1}{2\pi}\int_{0}^{2\pi}f(\theta)\overline{g(\theta)}\,d\theta. $$ You can see this because every $f \in L^{2}[0,2\pi]$ can be written as the orthogonal sum $$ f = \sum_{n=-\infty}^{\infty}(f,e^{in\theta})e^{in\theta}. $$ The unilateral shift is multiplication by $z$ on the Hardy space $H^{2}(D)$ consisting of all holomorphic functions $f(z)=\sum_{n=0}^{\infty}a_n z^{n}$ in the unit disk $D$ such that $$ \|f\|^{2} = \sum_{n=0}^{\infty}|a_n|^{2} < \infty. $$ Another common type of shift is a weighted shift where $$ e_{n} \mapsto \lambda_n e_{n+1}. $$