Shiftoperator

Shift operators are considered in the mathematical branch of functional analysis. When unilateral shift operator ( see below) there is a specific non-normal operator on a Hilbert space. This operator has many features for which there is no finite-dimensional representation.

Definition

An infinite-dimensional separable Hilbert space is by the theorem of Fischer- Riesz isometrically isomorphic to, with a countably infinite set is, for example, or. The operator

Ie bilateral shift operator.

Ie unilateral shift operator. The term shift operator stems from the fact that these operators shift the terms of the sequence to an index position. At the bilateral shift operator indices are affected on both sides of zero, positive and negative, the unilateral shift operator only the indexes of a page, ie only the positive. In the mathematical literature Shift operator is, without further addition, usually for the unilateral shift operator. Often you can also continue the verbal component operator and simply speaks of the shift.

Summing as a subspace of that by identifying, one sees that, that is the unilateral shift operator is a restriction of the bilateral shift operator.

The bilateral shift

The bilateral shift is a unitary operator, the reverse is the adjoint operator

.

The spectrum of the bilateral shift is the entire circle line, that is. No element of the spectrum is an eigenvalue.

The unilateral shift

The unilateral shift is an isometry which is not surjective because the image is the set of all sequences of whose first component is 0. Thus, an injective linear operator that is not injective; This is a phenomenon which does not occur in the theory of finite areas, that is in linear algebra.

The adjoint operator is

.

This immediately follows and, where the latter stands for the orthogonal projection onto the image of. In particular, is not normal. One can even show that the shift operator of any unitary operator has exactly the maximum possible norm distance 2.

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 ​​, ie there is a sequence of vectors with norm 1, so that. For the inner spectral points with this is not true.

The spectrum of the adjoint operator is also the full disc and the edge of the circle is also made louder approximate eigenvalues ​​which are not real eigenvalues. The internal spectral points with are all eigenvalues ​​of. The corresponding eigenspaces are all one-dimensional, the eigenspace to be generated from.

The shift operator as a Fredholm operator

The shift operator is a Fredholm operator with. Therefore, the image in the Calkin algebra is unitary, but you'd also without the concept of the Fredholm operator removes the formulas and. The spectrum of the circle.

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, so called a subspace invariant (with respect to ) if. One can now describe all isometries on a Hilbert space with these terms. An isometry is essentially a direct sum of a unitary operator and some shift operators, more precisely:

  • If an isometry on a Hilbert space, then decays into a direct sum of invariant subspaces, so that unitary and each operator is unitarily equivalent to the shift operator.

It may be, that is the proportion of unitary isometry disappears, but also, and thus, the isometry is unitary. This representation is called an isometry their Wold decomposition or Wold - von Neumann decomposition ( after Herman Wold and John von Neumann ).

The shift operator on H2

Be the circle line and normalized to 1 Lebesgue measure on, ie the size of the Lebesgue measure on the unit interval [0,1] under the mapping. , The so-called Hardy space is defined as the subspace generated by the functions in the Hilbert space.

It can be shown that the multiplication by the function defines a constant linear operator on. Since the functions form an orthogonal basis of the Hardy - space, this operator is unitarily equivalent to the shift operator, and it also refers to him simply as a shift operator. In this particular illustration of the shift operator of the shift operator appears as a multiplication operator.

Swell

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