Shift operator

A shift operator translates the components of a function or sequence by a fixed displacement. Bilateral shifts act on doubly infinite sequences; the unilateral shift operator acts on a one-sided sequence space and has distinct forward and backward versions.

The unilateral shifts on sequences move every component one place left or right. Their spectra illustrate behavior absent in finite dimensions.

On the Hilbert space ${\ell }^2(\mathbb{N})$, the left shift deletes the first coordinate. It is the adjoint operator of the unilateral shift operator $S$. The identities $LS=I$ and $SLf=f-f_1{e}_1$ show that it is also the Moore–Penrose inverse of an operator for $S$; both maps have operator norm one.

An operator $T$ is a unilateral shift exactly when it is an isometry, $(\mathrm{Im}T{)}^{\perp }$ is one-dimensional, and ${\bigcap }_{n\ge 1}\mathrm{Im}(T^n)=\{0\}$. A unit vector $e_1$ in the first orthogonal complement is wandering, and $e_n=T^{n-1}{e}_1$ is a complete orthonormal basis.