Self-adjoint operator
A self-adjoint operator is a linear operator with special properties. Operators, and in particular self-adjoint operators are investigated in the mathematical branch of functional analysis. The self-adjoint operator is a generalization of the self-adjoint matrix.
- 4.1 Symmetric Matrix
- 4.2 The operator -i d / dx
- 4.3 Laplacian
- 4.4 multiplication operator
- 5.1 The first criterion
- 5.2 The second criterion
- 5.3 Third criterion
- 5.4 Fourth criterion
- 8.1 spectral Decomposition
- 8.2 multiplication operator
Definition
Be a Hilbert space consisting of the vector space and the scalar product and be a densely defined operator. A densely defined operator is then called to adjoint if
Applies to all. With and the domains of the operators and designated. The adjoint operator is usually listed with a, so you shall.
An operator is now called self-adjoint if and only if, that matches the operator with its adjoint operator.
History
John von Neumann, who in 1929 founded the theory of unbounded operators also the first to recognize the need to distinguish between symmetric and self-adjoint operators. Only for the latter, a spectral decomposition, as described in the last section of this article, are shown. Von Neumann called symmetric Hermitian operators. He noted that it was important, among others, for the spectral decomposition that an operator does not allow for symmetric extension and called this class of operators maximum Hermitian. However, this requirement for the spectral theorem, which requires self-adjoint operators, are not sufficient. Von Neumann called self-adjoint on excitation Erhard Schmidt operators hyper maximal. The term self-adjoint operator was coined by Marshall Harvey Stone.
Related Items
Self-adjoint matrix
Be the real or complex number field and let a scalar product on then is a Hilbert space. A matrix is called self-adjoint if
Applies to all. The matrix is considered here as a linear map on the. Since maps between finite dimensional vector spaces is limited therefore defined continuous and thus also dense. So is a self-adjoint matrix, also a self-adjoint operator. Looking at the with its standard scalar product, the symmetric matrices correspond to self-adjoint. In the case of the corresponding canonical scalar product of the Hermitian matrices are self-adjoint.
Symmetric operator
An operator is called symmetric if
Applies to all. In contrast to the self-adjoint operator is not required here that the operator must be tightly defined ( but that is not uniform in the literature ). Is tightly defined (and thus the adjoint operator is well-defined ), so is symmetric if and only if. For bounded operators, the terms self-adjoint and symmetric fall together. Therefore, symmetric, non- self-adjoint operators are always unlimited. In addition, the set of Hellinger - Toeplitz states that every symmetric operator which is defined on all of, is continuous and self-adjoint with it.
Essential self-adjoint operator
An operator is called essentially self-adjoint, if symmetrical, densely defined and its seclusion is self-adjoint. A much self-adjoint operator so you can always continue to be a self-adjoint operator.
Examples
Symmetric Matrix
A symmetric matrix can be understood as an operator. Regarding the Standardskalarproduktes is any symmetric matrix is a self-adjoint matrix or a self-adjoint operator.
The operator -i d / dx
If an operator is bounded, then the terms symmetric operator, essentially self-adjoint and self-adjoint operator as mentioned equivalent. For unrestricted, although the self-adjoint operators implies the symmetry, but the converse is not true. A counterexample is the following pair:
From the chain of equations
It follows that the operators are symmetrical. However, only the operator is self-adjoint, because in the first case, the domain is restricted unnecessarily. He then has no more intrinsic functions, because these are all of the form, so would violate the required condition.
Laplace operator
The Laplacian operator is unrestricted. He is self-adjoint with respect to the scalar product. That is, it is symmetric with respect to this scalar product, which
For all, and is densely defined. The derivation is to be understood in the weak sense. Thus true for the domain
This corresponds to the Sobolev space of weakly differentiable functions quadratintegierbaren and twice, this is dense in. The symmetry of the Laplace operator follows from the Green 's formula.
Multiplication operator
Let be a measure space and a measurable function. The multiplication operator is defined by
This operator is defined unbounded and dense, because for contains all the classes that vanish outside and because is tight. In addition, the scalar product is symmetric with respect. The operator is also self-adjoint. Namely, since valid for a symmetric operator, and what means need only be shown for the self-adjoint. Be the characteristic function of, for and applies
That is true almost everywhere. Since converges pointwise, applies almost everywhere. Now, since in is what shows and thus proves the self-adjoint.
Criteria
For a densely defined operator in a Hilbert space there are the following repeatedly mentioned criteria related to the question of self-adjointness.
The first criterion
Is then and only then self-adjoint operator in if the following condition is met:
The second criterion
Is then and only then self-adjoint operator in if the following conditions are met:
For the null spaces occurring in the latter condition is often considered the Hilbert space dimensions. This is called in the case of a symmetric operator and its deficiency indices. The latter condition can therefore also be expressed by saying that the deficiency indices equal to 0.
Third criterion
The conditions 2 and 3 of the second criterion can be reinterpreted as a single and in this way one obtains with regard to the question of self-adjointness of a further equivalent criterion:
Is then and only then self-adjoint operator in if the following conditions are met:
Fourth criterion
The fourth criterion shows that the self-adjoint of a densely defined operator is essentially determined by the position of its spectrum within the real numbers:
Is then and only then self-adjoint operator in if the following conditions are met:
Properties
Be a densely defined operator on the Hilbert space
- Then a self-adjoint operator with
Be a self-adjoint operator on the Hilbert space
- It is the spectrum of there is no spectral values are true complex numbers. In particular, a self-adjoint matrix has only real eigenvalues or spectral.
- An operator is positive, that is to say it applies to all exactly when the inclusion applies to the spectrum.
- If true, then there exists a self-adjoint operator with so true.
Friedrichs extension
Be a Hilbert space and a densely defined semi- bounded operator. For an operator means to be a semi- confined, that the operator is either an inequality or inequality for one and for all fulfilled. Then there exists a self-adjoint extension of To, which satisfies the same estimate.
It should be noted that in a semi- bounded operator of the expression must be real-valued, since otherwise the order relations and not defined; and operators for which holds for all symmetric.
Let be a closed and densely defined operator. Then it can be concluded from the Friedrich 's extension that is densely defined and self-adjoint.
Spectral theorem for unbounded operators
Spectral
Be a Hilbert space and the Borel σ - algebra. For every self-adjoint operator a unique spectral measure exists, so that
Applies with and. This statement is the spectral theorem for unbounded self-adjoint operators. If one requires that the operators limited and self-adjoint or are even compact and self-adjoint, then simplifying the result. This is explained in the article Spectral closer.
Multiplication operator
Be also again a Hilbert space and be a self-adjoint operator. Then there exists a ( in the separable case, a - finite ) measure space, a measurable function and a unitary operator with
In essence, therefore, the multiplication operator is the only example of a self-adjoint operator.
Swell
- Hans Cycon, Richard G. Froese, Werner Kirsch, Barry Simon, Schrödinger Operators, Springer 1987
- Friedrich Hirzebruch / Winfried Scharlau: Introduction to Functional Analysis. ( = Set "BI - university paperbacks ", Volume No. 296 ). Bibliographical Institute, Mannheim [u a ] 1971, ISBN 3-411-00296-4. MR0463864
- Reinhold Meise - Dietmar Vogt: Introduction to Functional Analysis ( = Vieweg Studies - Advanced Mathematics 62. ). Vieweg Verlag, Braunschweig [ua ] 1992, ISBN 3-528-07262-8. MR1195130
- Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, 4 volumes, Academic Press 1978, 1980
- Walter Rudin: Functional Analysis. McGraw- Hill, New York 1991. ISBN 0070542368th chap. 13
- Gerald Teschl: Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators, American Mathematical Society, Providence RI, 2009. ISBN 978-0-8218-4660-5 ( free online version)
- Dirk Werner: Functional Analysis, Springer-Verlag. Berlin 2007, ISBN 978-3-540-72533-6. P 342 ff