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.

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 $S$:

• $\sigma (S^{*})={\overline {\mathbb {D} }}$ (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$$U(a_{0},a_{1},a_{2},\dots )=\sum _{n=0}^{\infty }a_{n}z^{n}$$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.

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.$$\{S\}'=\{M_{\varphi }:\varphi \in H^{\infty }\}$$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:

• 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.

The $S$-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 $M_{u}(H^{2}),M_{v}(H^{2})$, we have $M_{u}(H^{2})\subset M_{v}(H^{2})$ iff $u/v\in H^{2}$.

See also

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

References

• Garcia, Stephan Ramon; Mashreghi, Javad; Ross, William T. (2023). "5. The Unilateral Shift". Operator Theory by Example. Oxford University Press. pp. 109–132. doi:10.1093/oso/9780192863867.003.0005. ISBN 9780192863867.