Shift operator

An operator that depends on a parameter and acts in a set of mappings (where is an Abelian semi-group and is a set) in accordance with the formula

( is also called the operator of shift by ). The semi-group is often taken to be or (then is a shift in some space of functions of a real variable), or (then is a shift in some space of sequences). The set and the corresponding set 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).