Shift operator

This article covers the shift operator from mathematics. For the shift operators from computer science see bitwise operator .

Shift operators (shift operators, shift operators, shift operators) are considered in the mathematical sub-area of functional analysis. The unilateral shift operator (see below) is a concrete non- normal operator on a Hilbert space . This operator has many properties for which there is no finite-dimensional equivalent.

definition

An infinite-dimensional separable Hilbert space is isometrically isomorphic to according to the Fischer-Riesz theorem , where a countably infinite set is, for example or . The operator ${\displaystyle \textstyle \ell ^{2}(I):=\{(a_{i})_{i\in I}\in {\mathbb {C} }^{I};\,\sum _{i\in I}|a_{i}|^{2}<\infty \}}$ ${\displaystyle I}$ ${\displaystyle I={\mathbb {Z} }}$ ${\displaystyle I={\mathbb {N} }}$

${\displaystyle W:\ell ^{2}({\mathbb {Z} })\rightarrow \ell ^{2}({\mathbb {Z} }),\,\,(a_{n})_{n\in {\mathbb {Z} }}\mapsto (a_{n-1})_{n\in {\mathbb {Z} }}}$

is called a bilateral shift operator .

${\displaystyle S:\ell ^{2}({\mathbb {N} })\rightarrow \ell ^{2}({\mathbb {N} }),\,\,(a_{n})_{n\in {\mathbb {N} }}\mapsto (0,a_{1},a_{2},\ldots )}$

is called a unilateral shift operator . The term shift operator comes from the fact that these operators shift the terms of the sequence by an index position. With the bilateral shift operator, indices on both sides of zero are affected, positive as well as negative, with the unilateral shift operator only the indices on one side, namely only the positive ones. In the mathematical literature, shift operator usually stands for the unilateral shift operator without any further addition. Often the word operator is left out and simply speaks of the shift .

If one understands as a subspace of by identifying with , one sees that is, that is, the unilateral shift operator is a restriction of the bilateral shift operator. ${\displaystyle \ell ^{2}({\mathbb {N} })}$ ${\displaystyle \ell ^{2}({\mathbb {Z} })}$ ${\displaystyle (a_{1},a_{2},a_{3},\ldots )}$ ${\displaystyle (\ldots ,0,a_{1},a_{2},a_{3},\ldots )}$ ${\displaystyle S=W|_{\ell ^{2}({\mathbb {N} })}}$

The bilateral shift

The bilateral shift is a unitary operator , the inverse is the adjoint operator ${\displaystyle W}$

${\displaystyle W^{*}:\ell ^{2}({\mathbb {Z} })\rightarrow \ell ^{2}({\mathbb {Z} }),\,\,(a_{n})_{n\in {\mathbb {Z} }}\mapsto (a_{n+1})_{n\in {\mathbb {Z} }}}$.

The spectrum of the bilateral shift is the entire circular line, that is . No element of the spectrum is an eigenvalue . ${\displaystyle \sigma (W)=\{\lambda \in {\mathbb {C} };\,|\lambda |=1\}}$

The unilateral shift

The unilateral shift is an isometry that is not surjective , because the image is the set of all sequences whose first component is 0. This is an injective linear operator that is not surjective; this is a phenomenon that does not appear in the theory of finite-dimensional spaces, that is, in linear algebra . ${\displaystyle S}$ ${\displaystyle \ell ^{2}({\mathbb {N} })}$ ${\displaystyle S}$

The adjoint operator is

${\displaystyle S^{*}\colon \ell ^{2}({\mathbb {N} })\rightarrow \ell ^{2}({\mathbb {N} }),\,\,(a_{n})_{n\in {\mathbb {N} }}\mapsto (a_{2},a_{3},\ldots )}$.

This immediately follows and , the latter standing for the orthogonal projection onto the image of . In particular, is not normal . One can even show that the shift operator of every unitary operator has exactly the maximum possible standard distance 2. ${\displaystyle S^{*}S=id}$ ${\displaystyle SS^{*}=P_{im(S)}}$ ${\displaystyle S}$ ${\displaystyle S}$

The spectrum of the shift operator

The spectrum of the full disc: . None of the spectral points is an eigenvalue . However, the spectral points with are so-called approximate eigenvalues , that is, there is a sequence of vectors with norm 1, so that . This does not apply to the inner spectral points with . ${\displaystyle S}$ ${\displaystyle \sigma (S)=\{\lambda \in {\mathbb {C} };\,|\lambda |\leq 1\}}$ ${\displaystyle \lambda }$ ${\displaystyle |\lambda |=1}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle \|Sx_{n}-\lambda x_{n}\|\to 0}$ ${\displaystyle \lambda }$ ${\displaystyle |\lambda |<1}$

The spectrum of the adjoint operator is also the full disk and the edge of the circle also consists of only approximate eigenvalues ​​that are not real eigenvalues. The inner spectral points with are all eigenvalues ​​of . The associated eigenspaces are all one-dimensional, the eigenspace zu is generated by. ${\displaystyle S^{*}}$ ${\displaystyle \lambda }$ ${\displaystyle |\lambda |<1}$ ${\displaystyle S^{*}}$ ${\displaystyle \lambda }$ ${\displaystyle (1,\lambda ,\lambda ^{2},\lambda ^{3},\ldots )}$

The shift operator as a Fredholm operator

The shift operator is a Fredholm operator with . Therefore, the image in Calkin's algebra is unitary, which can be taken from the formulas and even without the term Fredholm operator . The spectrum of is the circular line. ${\displaystyle S}$ ${\displaystyle \mathrm {ind} (S)=\mathrm {dim} (\mathrm {ker} (S))-\mathrm {codim} (\mathrm {im} (S))=-1}$ ${\displaystyle \pi (S)}$ ${\displaystyle S^{*}S=id}$ ${\displaystyle SS^{*}=P_{im(S)}}$ ${\displaystyle \pi (S)}$

Wold decomposition

A continuous linear operator on a Hilbert space H is unitarily equivalent to the shift operator if there is a unitary operator with . If any operator is on , then a subspace is called invariant (with respect to ), if . With these terms one can now describe all isometries on a Hilbert space. An isometry is essentially a direct sum of a unitary operator and some shift operators, more precisely: ${\displaystyle T}$ ${\displaystyle U:H\rightarrow \ell ^{2}({\mathbb {N} })}$ ${\displaystyle T=U^{*}SU}$ ${\displaystyle T}$ ${\displaystyle H}$ ${\displaystyle V\subset H}$ ${\displaystyle T}$ ${\displaystyle T(V)\subset V}$

• If an isometry is on a Hilbert space , then it breaks down into a direct sum of invariant subspaces, so that is unitary and every operator is unitarily equivalent to the shift operator. ${\displaystyle T}$ ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle \textstyle G\oplus \bigoplus _{i\in I}H_{i}}$ ${\displaystyle T|_{G}}$ ${\displaystyle T|_{H_{i}}}$

It can be, that is, the unitary part of the isometry disappears, but also and thus , then the isometry is unitary. This representation of an isometry is also called its Wold decomposition or Wold von Neumann decomposition (after Herman Wold and John von Neumann ). ${\displaystyle G=\{0\}}$ ${\displaystyle G=H}$ ${\displaystyle I=\emptyset }$

The shift operator on H ²

Let the circle line and the Lebesgue measure normalized to 1 be on , that is, the image measure of the Lebesgue measure on the unit interval [0,1] below the figure . , the so-called Hardy space , is defined as the subspace in the Hilbert space generated by the functions . ${\displaystyle \Gamma =\{\lambda \in {\mathbb {C} };\,|\lambda |=1\}}$ ${\displaystyle \mu }$ ${\displaystyle \Gamma }$ ${\displaystyle t\to e^{2\pi it}}$ ${\displaystyle H^{2}}$ ${\displaystyle z\mapsto z^{n},n=0,1,2,\ldots }$ ${\displaystyle L^{2}(\Gamma ,\mu )}$

One can show that the multiplication with the function defines a continuous, linear operator on . Since the functions form an orthogonal basis of Hardy space, this operator is unitarily equivalent to the shift operator, and it is also referred to simply as the shift operator. In this particular representation of the shift operator, the shift operator appears as a multiplication operator. ${\displaystyle id_{\Gamma }:z\mapsto z}$ ${\displaystyle H^{2}}$ ${\displaystyle z\mapsto z^{n}}$

swell

• Paul Halmos : A Hilbert Space Problem Book, Springer-Verlag, ISBN 0387906851