In operator theory, the unilateral shift is a one-sided shift operator, that is, a shift operator acting on one-sided sequences or shift spaces. The term "operator" is used to draw contrast to finite-dimensional shift matrices. The term "unilateral" draws a distinction to the bilateral shift operator, of which the Baker's map is an example. Shift operators are commonly studied in the context of measure-preserving dynamical systems. In such general settings, the unilateral shift operator is usually called the transfer operator or the Frobenius-Peron operator; its inverse is the Koopman operator. The properties of shift operators depend very strongly on the topology of the spaces on which they act; for example, the Bernoulli shift famously has a discrete spectrum given by the Bernoulli polynomials when acting on the space of bounded smooth functions on the unit interval, but has a continuous spectrum (on the unit disk), when acting on the Hilbert space of square-integrable functions. When acting on a measure space, the eigenfunctions of shift operators are characteristically fractal in shape, often differentiable-nowhere or even continuous-nowhere. Eigenvalues on the unit circle are associated with unitary time evolution, while those inside the unit disk are conventionally identified with decaying modes in statistical systems. In quantum mechanics, the prototypical unilateral shift operator is the annihilation operator of the quantum harmonic oscillator; it's eigenfunctions correspond to coherent states. This article deals primarily with unilateral shifts acting on Hilbert space, specifically in two representations: as an operator on the sequence space ℓ 2 {\displaystyle \ell ^{2}} , or as a multiplication operator on a Hardy space. Its properties, particularly its invariant subspaces, are well-understood and serve as a model for more general theories.
Definition Let ℓ 2 {\displaystyle \ell ^{2}} be the Hilbert space of square-summable sequences of complex numbers, i.e., ℓ 2 = { ( a 0 , a 1 , a 2 , … ) : a n ∈ C and ∑ n = 0 ∞ | a n | 2 < ∞ } {\displaystyle \ell ^{2}=\left\{(a_{0},a_{1},a_{2},\dots ):a_{n}\in \mathbb {C} {\text{ and }}\sum _{n=0}^{\infty }|a_{n}|^{2}<\infty \right\}} The unilateral shift is the linear operator S : ℓ 2 → ℓ 2 {\displaystyle S:\ell ^{2}\to \ell ^{2}} defined by:
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