Inmathematics, and in particularfunctional analysis, theshift operator, also known as thetranslation operator, is anoperator that takes afunctionx ↦f(x)to itstranslationx ↦f(x +a).^[1] Intime series analysis, the shift operator is called thelag operator.

Shift operators are examples oflinear operators, important for their simplicity and natural occurrence. The shift operator action onfunctions of a real variable plays an important role inharmonic analysis, for example, it appears in the definitions ofalmost periodic functions,positive-definite functions,derivatives, andconvolution.^[2] Shifts of sequences (functions of aninteger variable) appear in diverse areas such asHardy spaces, the theory ofabelian varieties, and the theory ofsymbolic dynamics, for which thebaker's map is an explicit representation. The notion oftriangulated category is a categorified analogue of the shift operator.

The shift operatorT ^t (where⁠${\displaystyle t\in \mathbb{R}}$⁠) takes a functionf on⁠${\displaystyle \mathbb{R}}$⁠ to its translationf_t,

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

A practicaloperational calculus representation of the linear operatorT ^t in terms of the plain derivative⁠${\displaystyle \frac{d}{dx}}$⁠ was introduced byLagrange,

which may be interpreted operationally through its formalTaylor expansion int; and whose action on the monomialxⁿ is evident by thebinomial theorem, and hence onall series inx, and so all functionsf(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 celebratedadvective 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 coordinatesh (Abel functions) are defined such that

${\displaystyle h^{\prime }(x)\equiv \frac{1}{\beta (x)},f(x)\equiv F(h(x)).}$

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

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

${\displaystyle \beta (x)=\frac{1}{x}}$ 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

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

etc.

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

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

Theleft shift operator acts on one-sidedinfinite sequence of numbers by

${\displaystyle S^{\ast }:(a_1,a_2,a_3,\dots )\mapsto (a_2,a_3,a_4,\dots )}$

and on two-sided infinite sequences by

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

Theright shift operator acts on one-sidedinfinite sequence of numbers by

${\displaystyle S:(a_1,a_2,a_3,\dots )\mapsto (0,a_1,a_2,\dots )}$

and on two-sided infinite sequences by

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

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

In general, as illustrated above, ifF is a function on anabelian groupG, andh is an element ofG, the shift operatorT ^g mapsF 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 standardnorms which appear in functional analysis. Therefore, it is usually acontinuous operator with norm one.

The shift operator acting on two-sided sequences is aunitary 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:$${\displaystyle \mathcal{F}{T}^t=M^t\mathcal{F},}$$whereM ^t is themultiplication operator byexp(itx). Therefore, the spectrum ofT ^t is theunit circle.

The one-sided shiftS acting on⁠${\displaystyle {\ell }_2(\mathbb{N})}$⁠ is a properisometry withrange equal to allvectors which vanish in the firstcoordinate. The operatorS is acompression ofT⁻¹, in the sense that$${\displaystyle T^{-1}y=Sx\text{ for each }x\in {\ell }^2(\mathbb{N}),}$$wherey is the vector in⁠${\displaystyle {\ell }_2(\mathbb{Z})}$⁠ withy_i =x_i fori ≥ 0 andy_i = 0 fori < 0. This observation is at the heart of the construction of manyunitary dilations of isometries.

Thespectrum ofS is theunit disk. The shiftS is one example of aFredholm operator; it has Fredholm index −1.

Jean Delsarte introduced the notion ofgeneralized shift operator (also calledgeneralized displacement operator); it was further developed byBoris Levitan.^[2][8][9]

A family of operators⁠${\displaystyle \{L^x{\}}_{x\in X}}$⁠ acting on a spaceΦ of functions from a setX 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 existse inX such thatL^e is theidentity operator.

In this case, the setX is called ahypergroup.

• 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.

• Unitary operators

• Articles with short description
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