Shift operator

Functions of a real variable

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]

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

Sequences

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.

Abelian groups

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]

${\displaystyle F_{g}(h)=F(h+g).}$

Action on Hilbert spaces

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.