The shift operator T ^t (where ⁠${\displaystyle t\in \mathbb {R} }$ ⁠) takes a function f on ⁠${\displaystyle \mathbb {R} }$ ⁠ to its translation f_t,

${\displaystyle T^{t}f(x)=f_{t}(x)=f(x+t)~.}$ 

A practical operational calculus representation of the linear operator T ^t in terms of the plain derivative ⁠${\displaystyle {\tfrac {d}{dx}}}$ ⁠ was introduced by Lagrange,

which may be interpreted operationally through its formal Taylor expansion in t; and whose action on the monomial xⁿ is evident by the binomial theorem, and hence on all series in x, and so all functions f(x) as above.^[3] This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.

The operator thus provides the prototype^[4] for Lie's celebrated advective flow for Abelian groups,

${\displaystyle \exp \left(t\beta (x){\frac {d}{dx}}\right)f(x)=\exp \left(t{\frac {d}{dh}}\right)F(h)=F(h+t)=f\left(h^{-1}(h(x)+t)\right),}$ 

where the canonical coordinates h (Abel functions) are defined such that

${\displaystyle h'(x)\equiv {\frac {1}{\beta (x)}}~,\qquad f(x)\equiv F(h(x)).}$ 

For example, it easily follows that ${\displaystyle \beta (x)=x}$  yields scaling,

${\displaystyle \exp \left(tx{\frac {d}{dx}}\right)f(x)=f(e^{t}x),}$ 

hence ${\displaystyle \exp \left(i\pi x{\tfrac {d}{dx}}\right)f(x)=f(-x)}$  (parity); likewise, ${\displaystyle \beta (x)=x^{2}}$  yields^[5]

${\displaystyle \exp \left(tx^{2}{\frac {d}{dx}}\right)f(x)=f\left({\frac {x}{1-tx}}\right),}$ 

${\displaystyle \beta (x)={\tfrac {1}{x}}}$  yields

${\displaystyle \exp \left({\frac {t}{x}}{\frac {d}{dx}}\right)f(x)=f\left({\sqrt {x^{2}+2t}}\right),}$ 

${\displaystyle \beta (x)=e^{x}}$  yields

${\displaystyle \exp \left(te^{x}{\frac {d}{dx}}\right)f(x)=f\left(\ln \left({\frac {1}{e^{-x}-t}}\right)\right),}$ 

etc.

The initial condition of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equation^[6]

${\displaystyle f_{t}(f_{\tau }(x))=f_{t+\tau }(x).}$ 

The left shift operator acts on one-sided infinite sequence of numbers by

${\displaystyle S^{*}:(a_{1},a_{2},a_{3},\ldots )\mapsto (a_{2},a_{3},a_{4},\ldots )}$ 

and on two-sided infinite sequences by

${\displaystyle T:(a_{k})_{k\,=\,-\infty }^{\infty }\mapsto (a_{k+1})_{k\,=\,-\infty }^{\infty }.}$ 

The right shift operator acts on one-sided infinite sequence of numbers by

${\displaystyle S:(a_{1},a_{2},a_{3},\ldots )\mapsto (0,a_{1},a_{2},\ldots )}$ 

and on two-sided infinite sequences by

${\displaystyle T^{-1}:(a_{k})_{k\,=\,-\infty }^{\infty }\mapsto (a_{k-1})_{k\,=\,-\infty }^{\infty }.}$ 

The right and left shift operators acting on two-sided infinite sequences are called bilateral shifts.

In general, as illustrated above, if F is a function on an abelian group G, and h is an element of G, the shift operator T ^g maps F to^[6][7]

${\displaystyle F_{g}(h)=F(h+g).}$ 

The shift operator acting on real- or complex-valued functions or sequences is a linear operator which preserves most of the standard norms which appear in functional analysis. Therefore, it is usually a continuous operator with norm one.

The shift operator acting on two-sided sequences is a unitary operator on ⁠${\displaystyle \ell _{2}(\mathbb {Z} ).}$ ⁠ The shift operator acting on functions of a real variable is a unitary operator on ⁠${\displaystyle L_{2}(\mathbb {R} ).}$ ⁠

In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform: $${\displaystyle {\mathcal {F}}T^{t}=M^{t}{\mathcal {F}},}$$  where M ^t is the multiplication operator by exp(itx). Therefore, the spectrum of T ^t is the unit circle.

The one-sided shift S acting on ⁠${\displaystyle \ell _{2}(\mathbb {N} )}$ ⁠ is a proper isometry with range equal to all vectors which vanish in the first coordinate. The operator S is a compression of T⁻¹, in the sense that $${\displaystyle T^{-1}y=Sx{\text{ for each }}x\in \ell ^{2}(\mathbb {N} ),}$$  where y is the vector in ⁠${\displaystyle \ell _{2}(\mathbb {Z} )}$ ⁠ with y_i = x_i for i ≥ 0 and y_i = 0 for i < 0. This observation is at the heart of the construction of many unitary dilations of isometries.

The spectrum of S is the unit disk. The shift S is one example of a Fredholm operator; it has Fredholm index −1.

Jean Delsarte introduced the notion of generalized shift operator (also called generalized displacement operator); it was further developed by Boris Levitan.^[2][8][9]

A family of operators ⁠${\displaystyle \{L^{x}\}_{x\in X}}$ ⁠ acting on a space Φ of functions from a set X to ⁠${\displaystyle \mathbb {C} }$ ⁠ is called a family of generalized shift operators if the following properties hold:

1. Associativity: let ${\displaystyle (R^{y}f)(x)=(L^{x}f)(y).}$  Then ${\displaystyle L^{x}R^{y}=R^{y}L^{x}.}$ 
2. There exists e in X such that L^e is the identity operator.

In this case, the set X is called a hypergroup.

• Arithmetic shift
• Logical shift
• Clock and shift matrices
• Finite difference
• Translation operator (quantum mechanics)

• Partington, Jonathan R. (March 15, 2004). Linear Operators and Linear Systems. Cambridge University Press. doi:10.1017/cbo9780511616693. ISBN 978-0-521-83734-7.
• Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.