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 ${\displaystyle \ell ^{2}}$, 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.^[1]

Definition

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

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

Properties

Adjoint operator

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

Basic properties

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

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

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

Spectral theory

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

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

Proof

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

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

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

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

Hardy space model

The unilateral shift can be studied using complex analysis.

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

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

Commutant

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

Cyclic vectors

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

• For the unilateral shift ${\displaystyle S}$ on ${\displaystyle H^{2}}$, the cyclic vectors are the outer functions.
• A function ${\displaystyle f\in H^{2}}$ that has a zero in the open unit disk ${\displaystyle \mathbb {D} }$ 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 ${\displaystyle f\in H^{2}}$ that is bounded away from zero (i.e., ${\displaystyle \inf _{z\in \mathbb {D} }|f(z)|>0}$) is a cyclic vector.
• A function ${\displaystyle f\in H^{2}}$, that is in the open unit disk ${\displaystyle \mathbb {D} }$ is nonzero but ${\displaystyle \inf _{z\in \mathbb {D} }|f(z)|=0}$, may or may not be cyclic. For example, ${\displaystyle f(z)=1-z}$ is a cyclic vector.

The cyclic vectors are precisely the outer functions.

Lattice of invariant subspaces

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

The ${\displaystyle S}$-invariant subspaces make up a lattice of subspaces. The two lattice operators, join and meet, correspond to operations on inner functions.

Given two invariant subspaces ${\displaystyle M_{u}(H^{2}),M_{v}(H^{2})}$, we have ${\displaystyle M_{u}(H^{2})\subset M_{v}(H^{2})}$ iff ${\displaystyle u/v\in H^{2}}$.^{[1]: Sec. 5.8}

See also

• Bilateral shift
• Hardy space
• H²
• Operator theory
• Beurling's theorem
• Toeplitz operator