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 ${\displaystyle \ell ^{2}}${\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] [2]}

Let ${\displaystyle \ell ^{2}}${\displaystyle \ell ^{2}} be the Hilbert space of square-summable sequences of complex numbers, i.e., $${\displaystyle \ell ^{2}=\left\{(a_{0},a_{1},a_{2},\dots ):a_{n}\in \mathbb {C} {\text{ and }}\sum _{n=0}^{\infty }|a_{n}|^{2}<\infty \right\}}$${\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 ${\displaystyle S:\ell ^{2}\to \ell ^{2}}${\displaystyle S:\ell ^{2}\to \ell ^{2}} defined by: $${\displaystyle S(a_{0},a_{1},a_{2},\dots )=(0,a_{0},a_{1},a_{2},\dots )}$${\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 ${\displaystyle (e_{n})_{n=0}^{\infty }}${\displaystyle (e_{n})_{n=0}^{\infty }} for ${\displaystyle \ell ^{2}}${\displaystyle \ell ^{2}}, where ${\displaystyle e_{n}}${\displaystyle e_{n}} is the sequence with a 1 in the n-th position and 0 elsewhere, the action of ${\displaystyle S}${\displaystyle S} is ${\displaystyle Se_{n}=e_{n+1}}${\displaystyle Se_{n}=e_{n+1}}. Its matrix representation is:$${\displaystyle S={\begin{bmatrix}0&0&0&0&\cdots \1円&0&0&0&\cdots \0円&1&0&0&\cdots \0円&0&1&0&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}}}$${\displaystyle S={\begin{bmatrix}0&0&0&0&\cdots \1円&0&0&0&\cdots \0円&1&0&0&\cdots \0円&0&1&0&\cdots \\\vdots &\vdots &\vdots &\vdots &\ddots \end{bmatrix}}}This is a Toeplitz operator whose symbol is the function ${\displaystyle f(z)=z}${\displaystyle f(z)=z}. It can be regarded as an infinite-dimensional lower shift matrix.

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

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

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

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

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

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

The unilateral shift can be studied using complex analysis.

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

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

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

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

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

The cyclic vectors are precisely the outer functions.

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

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

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

• 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.

• Operator theory
• Linear operators
• Hilbert spaces

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