Unilateral shift operator

In operator theory, the unilateral shift is an operator on a Hilbert space. It is often studied in two main representations: as an operator on the sequence space $\ell ^{2}$ , 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.^[1]

Definition

Let $\ell ^{2}$ be the Hilbert space of square-summable sequences of complex numbers, i.e., $\ell ^{2}=\left\{(a_{0},a_{1},a_{2},\dots ):a_{n}\in \mathbb {C} {\text{ and }}\sum _{n=0}^{\infty }|a_{n}|^{2}<\infty \right\}$ The unilateral shift is the linear operator $S:\ell ^{2}\to \ell ^{2}$ defined by: $S(a_{0},a_{1},a_{2},\dots )=(0,a_{0},a_{1},a_{2},\dots )$ This operator is also called the forward shift.

With respect to the standard orthonormal basis $(e_{n})_{n=0}^{\infty }$ for $\ell ^{2}$ , where $e_{n}$ is the sequence with a 1 in the n-th position and 0 elsewhere, the action of $S$ is $Se_{n}=e_{n+1}$ . Its matrix representation is: $S={\begin{bmatrix}0&0&0&0&\cdots \\1&0&0&0&\cdots \\0&1&0&0&\cdots \\0&0&1&0&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}}$ This is a Toeplitz operator whose symbol is the function $f(z)=z$ . It can be regarded as an infinite-dimensional lower shift matrix.

Properties

Adjoint operator

The adjoint of the unilateral shift, denoted $S^{*}$ , is the backward shift. It acts on $\ell ^{2}$ as: $S^{*}(b_{0},b_{1},b_{2},b_{3},\dots )=(b_{1},b_{2},b_{3},\dots )$ The matrix representation of $S^{*}$ is the conjugate transpose of the matrix for $S$ : $S^{*}={\begin{bmatrix}0&1&0&0&\cdots \\0&0&1&0&\cdots \\0&0&0&1&\cdots \\0&0&0&0&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}}$ It can be regarded as an infinite-dimensional upper shift matrix.

Basic properties

$S,S^{*}$ are both continuous but not compact.
$S^{*}S=I$ .
$S,S^{*}$ make up a pair of unitary equivalence between $\ell ^{2}$ and the set of $\ell ^{2}$ -sequences whose first element is zero.

The resolvent operator has matrix representation $(zI-S)^{-1}={\begin{bmatrix}z^{-1}&0&0&0&\cdots \\z^{-2}&z^{-1}&0&0&\cdots \\z^{-3}&z^{-2}&z^{-1}&0&\cdots \\z^{-4}&z^{-3}&z^{-2}&z^{-1}&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}}$ which is bounded iff $|z|>1$ . Similarly, $(zI-S^{*})^{-1}=((z^{*}I-S)^{-1})^{*}$ .

For any $z\in \mathbb {C} ,a\in \ell ^{2}$ with $\|a\|=1$ , $\|(zI-S)a\|^{2}=1+|z|^{2}-2\Re (\langle Sa,a\rangle z),\quad \|(zI-S^{*})a\|^{2}=1-|a_{0}|^{2}+|z|^{2}-2\Re (\langle Sa,a\rangle z^{*})$ where $\Re$ is the real part.

Spectral theory

Spectrum of the forward shift—Let $\mathbb {D}$ be the open unit disk, ${\overline {\mathbb {D} }}$ the closed unit disk, and $\mathbb {T}$ the unit circle.

The spectrum of $S$ is $\sigma (S)={\overline {\mathbb {D} }}$ .
The point spectrum of $S$ is empty: $\sigma _{p}(S)=\emptyset$ .
The approximate point spectrum of $S$ is the unit circle: $\sigma _{ap}(S)=\mathbb {T}$ .

Proof

To show $\sigma (S)={\overline {\mathbb {D} }}$ , use the matrix representation of $(zI-S)^{-1}$ , and note that it is bounded iff $|z|>1$ . To show $\sigma _{p}(S)=\emptyset$ , directly show that $Sa=\lambda a$ implies $a=0$ .

To show $\sigma _{ap}(S)=\mathbb {T}$ , note that $\|(zI-S)a\|^{2}\geq 1+|z|^{2}-2|z|=(1-|z|)^{2}$ for any $z\in \mathbb {C} ,a\in \ell ^{2}$ with $\|a\|=1$ , so $\sigma _{ap}(S)\subset {\overline {\mathbb {D} }}\setminus \mathbb {D} =\mathbb {T}$ . Conversely, for any $z\in \mathbb {T}$ , construct the following unit vector $a={\frac {1}{\sqrt {N}}}(1,z^{-1},z^{-2},\dots ,z^{-(N-1)},0,0,\dots )$ then $\|(zI-S)a\|^{2}=2/N$ , which converges to 0 at $N\to \infty$ .

The spectral properties of $S^{*}$ differ significantly from those of $S$ :^[1]^{: Proposition 5.2.4}

$\sigma (S^{*})={\overline {\mathbb {D} }}$ (since $\sigma (A^{*})={\overline {\sigma (A)}}$ ).
The point spectrum $\sigma _{p}(S^{*})$ is the entire open unit disk $\mathbb {D}$ . For any $\lambda \in \mathbb {D}$ , the corresponding eigenvector is the geometric sequence $(1,\lambda ,\lambda ^{2},\lambda ^{3},\dots )$ .
The approximate point spectrum $\sigma _{ap}(S^{*})$ is the entire closed unit disk ${\overline {\mathbb {D} }}$ . To show this, it remains to show $\mathbb {T} \subset \sigma _{ap}(S^{*})$ , which can be proven by a similar construction as before, using $a={\frac {1}{\sqrt {N}}}(1,z^{1},z^{2},\dots ,z^{(N-1)},0,0,\dots )$ .

Hardy space model

The unilateral shift can be studied using complex analysis.

Define the Hardy space $H^{2}$ as the Hilbert space of analytic functions $f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$ on the open unit disk $\mathbb {D}$ for which the sequence of coefficients $(a_{n})$ is in $\ell ^{2}$ .

Define the multiplication operator $M_{z}$ on $H^{2}$ : $(M_{z}f)(z)=zf(z)$ then $S$ and $M_{z}$ are unitarily equivalent via the unitary map $U:\ell ^{2}\to H^{2}$ defined by^[1] $U(a_{0},a_{1},a_{2},\dots )=\sum _{n=0}^{\infty }a_{n}z^{n}$ which gives $U^{*}M_{z}U=S$ . Using this unitary equivalence, it is common in the literature to use $S$ to denote $M_{z}$ and to treat $H^{2}$ as the primary setting for the unilateral shift.^[1]^{: Sec. 5.3}

Commutant

The commutant of an operator $A$ , denoted $\{A\}'$ , is the algebra of all bounded operators that commute with $A$ . The commutant of the unilateral shift is the algebra of multiplication operators on $H^{2}$ by bounded analytic functions.^[1]^{: Corollary 5.6.2} $\{S\}'=\{M_{\varphi }:\varphi \in H^{\infty }\}$ Here, $H^{\infty }$ is the space of bounded analytic functions on $\mathbb {D}$ , and $(M_{\varphi }f)(z)=\varphi (z)f(z)$ .

Cyclic vectors

A vector $x$ is a cyclic vector for an operator $A$ if the linear span of its orbit $\{A^{n}x:n\geq 0\}$ is dense in the space. We have:^[1]^{: Sec. 5.7}

For the unilateral shift $S$ on $H^{2}$ , the cyclic vectors are the outer functions.
A function $f\in H^{2}$ that has a zero in the open unit disk $\mathbb {D}$ is not a cyclic vector. This is because every function in the span of its orbit will also be zero at that point, so the subspace cannot be dense.
A function $f\in H^{2}$ that is bounded away from zero (i.e., $\inf _{z\in \mathbb {D} }|f(z)|>0$ ) is a cyclic vector.
A function $f\in H^{2}$ , that is in the open unit disk $\mathbb {D}$ is nonzero but $\inf _{z\in \mathbb {D} }|f(z)|=0$ , may or may not be cyclic. For example, $f(z)=1-z$ is a cyclic vector.

The cyclic vectors are precisely the outer functions.

Lattice of invariant subspaces

The $S$ -invariant subspaces of $H^{2}$ are completely characterized analytically. Specifically, they are precisely $M_{u}(H^{2})$ where $u$ is an inner function.