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In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. They are used in diverse areas, from Hardy spaces to the theory of abelian varieties. (There is another usage of shift operator as a translation operator : see for example Sheffer sequence.)

A typical one-sided shift operator takes an infinite sequence of numbers

(a₁, a₂, ...)

to

(0, a₁, a₂, ...).

This operation respects typical convergence conditions, such as absolute convergence of the corresponding infinite series; it therefore gives rise to continuous operators on the standard sequence spaces used in functional analysis, usually with norm 1.

Another way to look at it would be in terms of polynomials: the sequences that eventually end in a string

(..., 0, 0, 0, ...)

or, in other words, having only a finite number of non-zero entries, are in a 1-1 correspondence with polynomials in an indeterminate T having a_i as coefficient of Tⁱ. The advantage of this representation is then that the shift operator becomes multiplication by T: this reveals quickly several aspects of its structure. Spaces of polynomials carry numerous topological structures; shift operators can be constructed by extensionFunctional analysis In functional analysis, it is often convenient to define something on a normed vector space by defining it on a dense set and extending it to the whole space. This procedure is justified for bounded linear operators by the theorem belo on corresponding complete spaces.

The bilateral shift operators are the related operators in which the sequences are bi-infinite (functions on the integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which sts, rather than just the natural numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("thiss). One can say that the analogue in this case of the polynomial representation is that by Laurent polynomials. The theory of analytic functionIn mathematics, an analytic function is one that is locally given by a convergent power series. Complex analysis teaches us that if a function f of one complex variable is differentiable in some open disk D centered at a point c in the complex field, thens is related to that of polynomials, by allowing infinite power seriesIn mathematics, a power series (in one variable) is an infinite series of the form :: where the coefficients a the center a and the argument x are usually real or complex numbers. These series usually arise as the Taylor series of some known function; the; on the other hand meromorphic functionA meromorphic function is a function that is holomorphic everywhere on the complex plane (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. Every meromorphic function cans have Laurent series that terminate in the direction of negative exponents. In the same way, the one-sided and bilateral shifts have rather different properties.