Spectral measure

In mathematics , especially in functional analysis , a spectral measure is a mapping that assigns orthogonal projections of a Hilbert space to certain subsets of a fixed set . Spectral measures are used to formulate results in spectral theory of linear operators, such as B. the spectral theorem for normal operators. In addition, the term, but with a different meaning, is used in stochastics .

definition

Let there be a measurement space , a real or complex Hilbert space , the Banach space of continuous linear operators on and the set of orthogonal projectors of . A spectral measure for the triple is a map with the following properties: ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle H}$ ${\displaystyle L(H)}$ ${\displaystyle H}$ ${\displaystyle P(H)}$ ${\displaystyle H}$ ${\displaystyle (X,{\mathcal {A}},H)}$ ${\displaystyle E\colon {\mathcal {A}}\rightarrow L(H)}$

1. It applies . It is the identity . ${\displaystyle E\left(X\right)=I}$ ${\displaystyle I\colon H\rightarrow H}$ ${\displaystyle H}$
2. For each is , ie is projector-worthy. ${\displaystyle \Omega \in {\mathcal {A}}}$ ${\displaystyle E(\Omega )\in P(H)}$ ${\displaystyle E}$
3. For all is with a complex or signed measure on . ${\displaystyle x,y\in H}$ ${\displaystyle E_{x,y}\colon {\mathcal {A}}\rightarrow \mathbb {K} }$ ${\displaystyle E_{x,y}(\Omega )=\langle E(\Omega )x,y\rangle }$ ${\displaystyle {\mathcal {A}}}$

The quadruple is called a spectral measure space . ${\displaystyle (X,{\mathcal {A}},H,E)}$

Often, the figure defined in this way is as a partition of unity (ger .: resolution of the identity referred to). It is common from a well -quality projector degree (ger .: projection-valued measure , often short PVM) to speak. ${\displaystyle E}$ ${\displaystyle I}$

If a topological space , its topology and its Borel algebra , then a spectral measure on which the Borel measurement space is based is called a Borel spectral measure . If special or , the Borel spectral measure is called a real or complex spectral measure. The carrier of a Borel spectral measure is as ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {B}}(X)}$ ${\displaystyle E}$ ${\displaystyle (X,{\mathcal {B}}(X))}$ ${\displaystyle X=\mathbb {R} }$ ${\displaystyle X=\mathbb {C} }$

${\displaystyle \mathrm {supp} (E)=X\setminus \bigcup \{O\in {\mathcal {O}}\,|\,E(O)=0\}}$

Are defined. This is the complement of the largest open subset of for which is. ${\displaystyle G}$ ${\displaystyle X}$ ${\displaystyle E(G)=0}$

properties

Let it be a spectral measure for the date . Then the following statements apply: ${\displaystyle E}$ ${\displaystyle (X,{\mathcal {A}},H)}$

• ${\displaystyle E(\varnothing )=0}$
• Modularity: It applies to everyone . ${\displaystyle E(\Omega _{1}\cup \Omega _{2})+E(\Omega _{1}\cap \Omega _{2})=E(\Omega _{1})+E(\Omega _{2})}$ ${\displaystyle \Omega _{1},\Omega _{2}\in {\mathcal {A}}}$
• Multiplicativity: It applies to everyone . In particular, the projectors and commutate with each other and the image of is perpendicular to the image of if applies. ${\displaystyle E(\Omega _{1}\cap \Omega _{2})=E(\Omega _{1})\,E(\Omega _{2})}$ ${\displaystyle \Omega _{1},\Omega _{2}\in {\mathcal {A}}}$ ${\displaystyle E(\Omega _{1})}$ ${\displaystyle E(\Omega _{2})}$ ${\displaystyle E(\Omega _{1})}$ ${\displaystyle E(\Omega _{2})}$ ${\displaystyle \Omega _{1}\cap \Omega _{2}=\varnothing }$

In particular, every spectral measure is a finite additive vector measure .

If one sets for , then applies to all because of the polarization identity ${\displaystyle E_{x}:=E_{x,x}}$ ${\displaystyle x\in H}$ ${\displaystyle x,y\in H}$

${\displaystyle E_{x,y}={\frac {1}{4}}\sum _{n=0}^{3}i^{n}E_{x+i^{n}y}}$

in the complex case or

${\displaystyle E_{x,y}={\text{Re}}{\bigg (}{\frac {1}{4}}\sum _{n=0}^{3}i^{n}E_{x+i^{n}y}{\bigg )}}$

in the real case. In particular, the dimensions are known when the dimensions are known, so that you often only work with these. ${\displaystyle E_{x,y}}$ ${\displaystyle E_{x}}$

Equivalent definition

The following characterization of spectral measures is often found in the literature as a definition. A mapping is a spectral measure if and only if ${\displaystyle E\colon {\mathcal {A}}\rightarrow L(H)}$

1. ${\displaystyle E\left(X\right)=I}$ applies
2. ${\displaystyle E}$ is projector quality and
3. for every sequence of measurable, pairwise disjoint sets ${\displaystyle (\Omega _{i})_{i\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle E{\bigg (}\biguplus _{i\in \mathbb {N} }\Omega _{i}{\bigg )}=\sum _{i=1}^{\infty }E(\Omega _{i})}$

in terms of the strong operator topology . This property is sometimes referred to as pointwise additivity. ${\displaystyle \sigma }$

The designation dismantling of the unit for can now be explained as follows. If there is a countable division of into measurable quantities, then applies ${\displaystyle E}$ ${\displaystyle (\Omega _{i})_{i\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle \mathrm {id} _{H}=E(X)=E{\bigg (}\biguplus _{i\in \mathbb {N} }\Omega _{i}{\bigg )}=\sum _{i=1}^{\infty }E(\Omega _{i})}$

or.

${\displaystyle H=\bigoplus _{i=1}^{\infty }(E(\Omega _{i})(H)),}$

where the orthogonal sum in the sense of Hilbert spaces is the family of closed subspaces. This corresponds to the fact that the eigenspaces of a normal operator form an orthogonal sum decomposition of . ${\displaystyle \bigoplus }$ ${\displaystyle \{E(\Omega _{i})(H)\,|\,i\in \mathbb {N} \}}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$

Examples

Let it be a normal linear operator. Then the spectrum of is not empty and consists of the eigenvalues of . The eigenspaces of the pairwise different eigenvalues ​​of are perpendicular to each other and have as an (inner) direct sum. This is equivalent to that ${\displaystyle A\colon \mathbb {C} ^{n}\rightarrow \mathbb {C} ^{n}}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle \mathbb {C} ^{n}}$

${\displaystyle \mathrm {id} _{\mathbb {C} ^{n}}=\sum _{\lambda \in \sigma (A)}P_{\lambda }}$

applies. It is the orthogonal projection of the eigenspace of the eigenvalue . From this representation of one obtains the ${\displaystyle P_{\lambda }}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle \operatorname {id} _{\mathbb {C} ^{n}}}$

${\displaystyle A=\sum _{\lambda \in \sigma (A)}\lambda \,P_{\lambda }}$

“Spectral resolution” of The spectral measure of is ${\displaystyle A.}$ ${\displaystyle A}$

${\displaystyle E(\Omega )=\sum _{\lambda \in \sigma (A)\cap \Omega }P_{\lambda }}$.

If any normal operator is , the spectrum of can be continuous or accumulate in a point and the above sum is replaced by a continuous summation term, namely by an (operator-valued) integral. ${\displaystyle A}$ ${\displaystyle A}$

• Every normal operator of a Hilbert space determines a spectral measure. According to the spectral theorem for normal operators, the operator is clearly described by this spectral measure. ${\displaystyle A}$ ${\displaystyle A}$
• Let L ² [0,1] be the Hilbert space of the square-summable functions in Lebesgue's sense on the unit interval and the Borel algebra of . For a substantially restricted function on denote the operator induced by multiplication by on . Describes the characteristic function for a Borel set of the unit interval and if one sets , then a spectral measure is defined for the tuple . This is the spectral measure of the multiplication operator . ${\displaystyle [0,1]}$ ${\displaystyle {\mathcal {B}}[0,1]}$ ${\displaystyle [0,1]}$ ${\displaystyle f}$ ${\displaystyle [0,1]}$ ${\displaystyle M_{f}}$ ${\displaystyle f}$ ${\displaystyle L^{2}[0,1]}$ ${\displaystyle \chi _{\Omega }}$ ${\displaystyle \Omega }$ ${\displaystyle E(\Omega ):=M_{\chi _{\Omega }}}$ ${\displaystyle E}$ ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),L^{2}[0,1])}$ ${\displaystyle M_{id}}$

Integration with respect to a spectral measure

Let it be a spectral measure space. Use the to associated complex measures there is certain - measurable functions a (usually unlimited) linear operator ${\displaystyle \left(X,{\mathcal {A}},H,E\right)}$ ${\displaystyle E}$ ${\displaystyle E_{x,y}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle f\colon X\rightarrow \mathbb {C} }$

${\displaystyle \int _{X}f\,dE}$

explain the Hilbert space . This operator is called the spectral integral of and the process by which it arises from is called the integration of with respect to the spectral measure. ${\displaystyle H}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle E}$

Spectral measure of a normal operator

Let it be a Hilbert space and a normal operator with a spectrum . Then one explains a spectral measure on the Borel algebra of . Let it be the functional calculus of the bounded Borel functions of . Since there is a morphism of -algebras, for every Borel set of the spectrum is given by an orthogonal projection of . One can show that is a spectral measure, the spectral measure of the normal operator . The spectral theorem for normal operators now says that ${\displaystyle H}$ ${\displaystyle A\in L(H)}$ ${\displaystyle \sigma \left(A\right)}$ ${\displaystyle E\colon {\mathcal {B}}(\sigma (A))\rightarrow L(H)}$ ${\displaystyle {\mathcal {B}}(\sigma (A))}$ ${\displaystyle \sigma \left(A\right)}$ ${\displaystyle \pi _{A}\colon {\mathcal {M}}_{\infty }(\sigma (A))\rightarrow L(H)}$ ${\displaystyle A}$ ${\displaystyle \pi _{A}}$ ${\displaystyle C^{*}}$ ${\displaystyle \Omega }$ ${\displaystyle A}$ ${\displaystyle E\left(\Omega \right):=\pi _{A}(\chi _{\Omega })}$ ${\displaystyle H}$ ${\displaystyle E\colon {\mathcal {B}}(\sigma (A))\rightarrow L(H)}$ ${\displaystyle A}$

${\displaystyle A=\int _{\sigma (A)}\lambda \,dE(\lambda )=\int _{\sigma (A)}id_{\sigma (A)}\,dE}$

applies. The spectral integral of the restricted Borel function with respect to the spectral measure is on the right-hand side of this equation . ${\displaystyle id_{\sigma \left(A\right)}}$ ${\displaystyle E}$

Spectral family

Definition of the spectral family

A family of orthogonal projectors is called a spectral family or spectral family if the following conditions are met: ${\displaystyle \{E_{\lambda }\,|\,\lambda \in \mathbb {\mathbb {R} } \}}$ ${\displaystyle E_{\lambda }\colon H\rightarrow H}$

• ${\displaystyle \lim _{\lambda \rightarrow -\infty }E_{\lambda }=0}$.
• ${\displaystyle \lim _{\lambda \rightarrow +\infty }E_{\lambda }=id_{H}}$.
• The family is continuous on the right , in the sense that applies. ${\displaystyle E}$ ${\displaystyle \lim _{\lambda \rightarrow \mu +}E_{\lambda }=E_{\mu }}$
• The family is growing monotonously : if it is, then it is . This condition is equivalent to the following condition: applies to all . ${\displaystyle E}$ ${\displaystyle \lambda \leq \mu }$ ${\displaystyle E_{\lambda }\leq E_{\mu }}$ ${\displaystyle \lambda ,\mu \in \mathbb {R} }$ ${\displaystyle E_{\lambda }E_{\mu }=E_{\mu }E_{\lambda }=E_{min\{\lambda ,\mu \}}}$

All occurring limits are to be considered in terms of the strong operator topology , i.e. point by point.

Relationship to the spectral measure

The concept of the spectral family historically preceded the concept of the spectral measure and was introduced by John von Neumann under the name Decomposition of the unit. The relationship between the two terms is given as follows: Each real spectral measure has exactly one spectral family and vice versa. The spectral measure and the spectral family determine each other through the relationship ${\displaystyle E}$ ${\displaystyle \{E_{\lambda }\,|\,\lambda \in \mathbb {\mathbb {R} } \}}$ ${\displaystyle E}$ ${\displaystyle \{E_{\lambda }\,|\,\lambda \in \mathbb {\mathbb {R} } \}}$

${\displaystyle E_{\lambda }=E((-\infty ,\lambda ])\quad ,\quad \lambda \in \mathbb {R} .}$

The carrier of the spectral family is the crowd ${\displaystyle \{E_{\lambda }\,|\,\lambda \in \mathbb {\mathbb {R} } \}}$

${\displaystyle {\overline {\{\lambda \in \mathbb {R} \,|\,E_{\lambda }\neq 0,E_{\lambda }\neq I\}}}.}$

With the help of a spectral family, the support of which is compact , one can, based on the Stieltjes integral, for a continuous function a, as ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }$

${\displaystyle \int _{-\infty }^{+\infty }f(\lambda )\,dE_{\lambda }}$

noted, define operator. This is clearly determined by the fact that he is the relationship

${\displaystyle \left\langle {\bigg (}\int _{-\infty }^{+\infty }f(\lambda )\,dE_{\lambda }{\bigg )}x,y\right\rangle =\int _{-\infty }^{+\infty }f(\lambda )\,d\langle E_{\lambda }x,y\rangle \quad ,\quad x,y\in H}$

fulfilled, with a conventional Stieltjes integral now on the right hand side . It then applies

${\displaystyle \int _{-\infty }^{+\infty }f(\lambda )\,dE_{\lambda }=\int _{\mathbb {R} }f(\lambda )\,dE(\lambda )}$,

if the corresponding spectral measure denotes. ${\displaystyle E}$ ${\displaystyle \{E_{\lambda }\,|\,\lambda \in \mathbb {\mathbb {R} } \}}$

Spectral measure of a bounded self-adjoint operator

The spectral family of a bounded self-adjoint operator has compact supports in , where ${\displaystyle [m,M]}$

${\displaystyle m=\inf _{||x||=1}\langle Ax,x\rangle }$

or.

${\displaystyle M=\sup _{||x||=1}\langle Ax,x\rangle }$

be. is sometimes referred to as spectral projection. The image of this orthogonal projection is imagined as a kind of generalized eigenspace. ${\displaystyle E_{\lambda }}$

Spectral measure of unlimited self-adjoint operators (quantum mechanics)

The measurable quantities of quantum mechanics correspond to (almost exclusively unrestricted, densely defined) essentially self-adjoint Hilbert space operators on separable Hilbert spaces ("observables", → mathematical structure of quantum mechanics ), with a spectral decomposition into three parts, in accordance with the above statements:

1. The first part is the point spectrum (the spectrum is countable; physicists misleadingly refer to it as "discrete"). Here you are dealing with sums .
2. The second part is the absolutely continuous spectrum (the spectrum is continuously uncountable; the physicists simply call it "continuous"). Ordinary integrals take the place of sums .
3. A singular, continuous spectral component is very rarely added (the spectrum is a Cantor set). Here one has to work with Stieltjes integrals (generated by non-differentiable monotonically increasing functions).

All observables show such a division and have the usual spectral dimensions and usual spectral projections. The aforementioned compactness of the spectrum does not apply.

The division into three parts, when weighted with the squares from the contributions of the eigenfunctions or the generalized eigenfunctions, results in exactly the value 1, in accordance with the probability interpretation of quantum mechanics.

In the case of a pure point spectrum, the spectral properties correspond to the postulate of the completeness of the eigenfunctions (expansion theorem) . In the case of an additional absolutely continuous spectral component, the physicists work, as mentioned, with so-called generalized eigenfunctions and wave packets (the connection with the spectral measure results from distribution theory using so-called Gelfand space triples ). A singular-continuous spectral component is usually not discussed at all, except e.g. B. in crystals with special "incommensurable" magnetic fields. More details in relevant textbooks on quantum mechanics and the measure theory of real functions.

literature

• John B. Conway : A Course in Functional Analysis (= Graduate Texts in Mathematics. Vol. 96). 2nd Edition. Springer, New York et al. 1990, ISBN 0-387-97245-5 .

• Paul R. Halmos : Introduction to Hilbert space and the theory of spectral multiplicity. Chelsea Publishing Company, New York NY 1951.
• Harro Heuser : Functional Analysis. Theory and application. 4th revised edition. BG Teubner, Wiesbaden 2006, ISBN 3-8351-0026-2 .
• Josef-Maria Jauch : Foundations of quantum mechanics. Addison-Wesley, Reading MA et al. 1968.
• Uwe Krey, Anthony Owen: Basic Theoretical Physics. A Concise Overview. Springer, Berlin et al. 2007, ISBN 978-3-540-36804-5 , special: Part III.
• Reinhold Meise, Dietmar Vogt: Introduction to functional analysis (= Vieweg studies. Vol. 62). Vieweg, Wiesbaden et al. 1992, ISBN 3-528-07262-8 .
• John von Neumann : General eigenvalue theory of Hermitean functional operators. In: Mathematical Annals . Vol. 102, No. 1, 1930, pp. 49-131, doi : 10.1007 / BF01782338 .
• Eduard Prugovečki : Quantum Mechanics in Hilbert Space. 2nd edition, Dover edition, unabridged republication. Dover Publications, Mineola NY 2006, ISBN 0-486-45327-8 .
• Dirk Werner : Functional Analysis. 5th enlarged edition. Springer, Berlin et al. 2005, ISBN 3-540-21381-3 .