# Shift operator

An operator $T _ {t}$ that depends on a parameter $t$ and acts in a set $\Phi$ of mappings $\phi : A \rightarrow E$( where $A$ is an Abelian semi-group and $E$ is a set) in accordance with the formula

$$T _ {t} \phi ( \cdot ) = \ \phi ( \cdot + t)$$

( $T _ {t}$ is also called the operator of shift by $t$). The semi-group $A$ is often taken to be $\mathbf R$ or $\mathbf R ^ {+}$( then $T _ {t}$ is a shift in some space of functions of a real variable), $\mathbf Z$ or $\mathbf N$( then $T _ {t}$ is a shift in some space of sequences). The set $E$ and the corresponding set $\Phi$ are usually endowed with a certain structure (of a vector, topological vector, normed, metric, or probability space).

A shift operator is used, in particular, in the theory of dynamical systems (see Shift dynamical system; Bernoulli automorphism). Also used is the terminology "shift operator along the trajectories of differential equations" (see Cauchy operator).