Lane class operator

The track class operators are examined in the mathematical discipline of functional analysis. They form an important class of linear operators on infinite-dimensional spaces. In contrast to general operators, some properties from the finite-dimensional case are retained with them, in particular their representation as a sum of one-dimensional operators. In important cases, the concept of track known from linear algebra is transferred to these operators, which is what gave them their name. In quantum mechanics , the track class operators appear as a density matrix .

Alexander Grothendieck's investigation of the theorem of the kernel from distribution theory also came across operators of the kind dealt with here and called them nuclear operators (Latin nucleus = kernel). This then led to the concept of nuclear space .

This article deals with the nuclear operators first on Hilbert spaces , then more generally on Banach spaces and finally on locally convex spaces .

motivation

Let be a vector space over the field of real or complex numbers. A one-dimensional operator is an operator of the form with and , where denotes the dual space of . In linear algebra , i.e. H. in this case , each linear mapping can be represented as a matrix with respect to a basis . For true even ${\displaystyle E}$ ${\displaystyle A:E\rightarrow E}$ ${\displaystyle A(x):=f(x)\cdot y}$ ${\displaystyle y\in E}$ ${\displaystyle f\in E'}$ ${\displaystyle E'}$ ${\displaystyle E}$ ${\displaystyle E={\mathbb {K} }^{n}}$ ${\displaystyle A:E\rightarrow E}$ ${\displaystyle (a_{i,j})_{i,j}}$ ${\displaystyle e_{1},\ldots e_{n}}$ ${\displaystyle x\in {\mathbb {K} }^{n}}$

${\displaystyle Ax=(a_{i,j})_{i,j}\cdot {\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}=\sum _{i=1}^{n}(a_{i,1},\ldots ,a_{i,n})\cdot {\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\cdot e_{i}}$.

${\displaystyle A}$is therefore a sum of one-dimensional operators. In order to be able to transfer this to infinite-dimensional spaces, one must form infinite sums of one-dimensional operators and therefore take precautions for their convergence. This leads to the following definition:

definition

Let and be two normalized vector spaces. An operator is called nuclear if there are sequences in and in with ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle A:E\rightarrow F}$ ${\displaystyle (a_{n})_{n}}$ ${\displaystyle F}$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle E\,'}$

${\displaystyle \sum _{n=1}^{\infty }\|a_{n}\|\cdot \|f_{n}\|<\infty }$

and

${\displaystyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)a_{n}}$

for everyone . Such a formula for is called a nuclear representation of . However, this is not clear. ${\displaystyle x\in E}$ ${\displaystyle A}$ ${\displaystyle A}$

The nuclear norm or trace norm of a nuclear operator is defined as

${\displaystyle \|A\|_{1}=\inf \sum _{n=1}^{\infty }\|a_{n}\|\|f_{n}\|,}$

where the infimum is formed over the sequences in and in , which result in a nuclear representation of . ${\displaystyle (a_{n})_{n}}$ ${\displaystyle F}$ ${\displaystyle (f_{n})_{n}}$ ${\displaystyle E\,'}$ ${\displaystyle A}$

Examples

• Be and be defined by . Then nuclear is with . In the Hilbert space case, equality applies. ${\displaystyle (a_{n})_{n}\in \ell ^{1}}$ ${\displaystyle A\colon \ell ^{p}\rightarrow \ell ^{p}}$ ${\displaystyle (x_{n})_{n}\mapsto (a_{n}x_{n})_{n}}$ ${\displaystyle A}$ ${\displaystyle \|A\|_{1}\leq \sum _{n}|a_{n}|}$ ${\displaystyle p=2}$
• Be continuous, be defined by . Then nuclear is with . ${\displaystyle k\colon [0,1]\times [0,1]\rightarrow {\mathbb {C} }}$ ${\displaystyle A\colon L^{\infty }[0,1]\rightarrow L^{\infty }[0,1]}$ ${\displaystyle (Af)(t):=\int _{0}^{1}k(t,s)f(s)\mathrm {d} s}$ ${\displaystyle A}$ ${\displaystyle \|A\|_{1}\leq \int _{0}^{1}\sup _{t}|k(t,s)|\mathrm {d} s}$
• Be defined by . Then there is a compact operator that is not nuclear. ${\displaystyle A\colon \ell ^{2}\rightarrow \ell ^{2}}$ ${\displaystyle (x_{n})_{n}\mapsto ({\tfrac {1}{n}}x_{n})_{n}}$ ${\displaystyle A}$

Simple properties

Let be the set of all nuclear operators . If complete , then with the nuclear norm there is a Banach space. The operators with a finite dimensional image are close together and every nuclear operator is compact . ${\displaystyle N(E,F)}$ ${\displaystyle E\to F}$ ${\displaystyle F}$ ${\displaystyle N(E,F)}$ ${\displaystyle E\to F}$ ${\displaystyle N(E,F)}$

The nuclear operators have called the Ideal Property : Be and normed spaces, is nuclear and and are continuous linear operators. Then is also nuclear and it is , where is the operator norm . It always applies ${\displaystyle E,\,F,\,G}$ ${\displaystyle H}$ ${\displaystyle A\colon E\to F}$ ${\displaystyle B\colon G\to E}$ ${\displaystyle C\colon F\to H}$ ${\displaystyle CAB\colon G\to H}$ ${\displaystyle \|CAB\|_{1}\leq \|C\|\|A\|_{1}\|B\|}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle \|\cdot \|\leq \|\cdot \|_{1}}$

Specifically, an ideal in algebra is the continuous linear operators , and with the nuclear norm is a Banach algebra . ${\displaystyle N(E):=N(E,E)}$ ${\displaystyle B(E)}$ ${\displaystyle E}$ ${\displaystyle N(E)}$

Nuclear operators on Hilbert spaces

In the Hilbert space the conditions are simpler. In these rooms, the nuclear operators were first examined in 1946 by Robert Schatten and John von Neumann . ${\displaystyle E=F=H}$

According to Fréchet-Riesz's theorem, each is of the form with one . A nuclear representation of an operator therefore has the form ${\displaystyle f_{n}\in H'}$ ${\displaystyle f_{n}(x)=\langle x,y_{n}\rangle }$ ${\displaystyle y_{n}\in H}$ ${\displaystyle A\colon H\rightarrow H}$

${\displaystyle A(x)=\sum _{n=1}^{\infty }\langle x,y_{n}\rangle x_{n}}$

with and ${\displaystyle x_{n},y_{n}\in H}$

${\displaystyle \sum _{n=1}^{\infty }\|x_{n}\|\|y_{n}\|<\infty .}$

If there is an arbitrary orthonormal basis of , then converges for each ${\displaystyle (e_{m})_{m}}$ ${\displaystyle H}$ ${\displaystyle A\in N(H)}$

${\displaystyle \sum _{m}\langle Ae_{m},e_{m}\rangle =\sum _{n}\langle x_{n},y_{n}\rangle }$,

where the left sum is to be read as the limit of the network of all finite partial sums in (i.e. as unconditional convergence ). This number is therefore independent of the choice of the orthonormal basis and also independent of the choice of the nuclear representation; it is called the trace of and is denoted by. Because of the English word trace for trace, the name is often found . ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle \mathrm {Sp} (A)}$ ${\displaystyle tr(A)}$

If is self-adjoint and is the sequence of the eigenvalues of counted with multiples , then and . For general the eigenvalue sequence is absolutely summable and it is . ${\displaystyle A\in N(H)}$ ${\displaystyle (\lambda _{n})_{n}}$ ${\displaystyle A}$ ${\displaystyle \textstyle \|A\|_{1}=\sum _{n}|\lambda _{n}|}$ ${\displaystyle \textstyle \mathrm {Sp} (A)=\sum _{n}\lambda _{n}}$ ${\displaystyle A\in N(H)}$ ${\displaystyle (\lambda _{n})_{n}}$ ${\displaystyle \textstyle \sum _{n}|\lambda _{n}|\leq \|A\|_{1}}$

As a further characterization one can show that an operator is nuclear if and only if it is the product of two Hilbert-Schmidt operators . ${\displaystyle A\in B(H)}$

${\displaystyle N(H)}$plays a central role in the duality theory of operator algebras . Denote the algebra of compact linear operators on . Each is defined by a continuous, linear functional . One can show that , is an isometric isomorphism, with the nuclear norm and the operator norm. In this sense, then . In the same way, each one defines a continuous linear functional through the formula and one can show again that , is an isometric isomorphism if one provides with the nuclear norm and with the operator norm. In this sense, then . In particular, is , that is, the spaces and are not reflexive for an infinitely dimensional Hilbert space . ${\displaystyle K(H)}$ ${\displaystyle H}$ ${\displaystyle A\in N(H)}$ ${\displaystyle \phi _{A}(T):=\mathrm {Sp} (AT)}$ ${\displaystyle K(H)}$ ${\displaystyle N(H)\rightarrow K(H)\,'}$ ${\displaystyle A\mapsto \phi _{A}}$ ${\displaystyle N(H)}$ ${\displaystyle K(H)}$ ${\displaystyle K(H)\,'\cong N(H)}$ ${\displaystyle T\in B(H)}$ ${\displaystyle \psi _{T}(A):=\mathrm {Sp} (AT)}$ ${\displaystyle N(H)}$ ${\displaystyle B(H)\rightarrow N(H)\,'}$ ${\displaystyle T\mapsto \psi _{T}}$ ${\displaystyle N(H)}$ ${\displaystyle B(H)}$ ${\displaystyle N(H)\,'\cong B(H)}$ ${\displaystyle K(H)\,''\cong B(H)}$ ${\displaystyle K(H),N(H)}$ ${\displaystyle B(H)}$

An analogy to sequence spaces

The following list contains an analogy between sequence spaces of complex numbers and operator algebras on a Hilbert space. In the sense of this analogy, one can consider the nuclear operators as a non-commutative version of the -sequences, it is at least a memory aid. ${\displaystyle \ell ^{1}}$

Episode space	Operator algebra
${\displaystyle c_{00}}$ = Space of finite sequences	${\displaystyle F(H)}$ = Algebra of operators of finite order
${\displaystyle c_{0}}$ = Space of zero sequences	${\displaystyle K(H)}$ = Algebra of compact operators
${\displaystyle \ell ^{1}}$ = Space of the absolute summable consequences	${\displaystyle N(H)}$ = Algebra of nuclear operators
${\displaystyle \ell ^{2}}$ = Space of the quadratically summable sequences	${\displaystyle HS(H)}$= Algebra of the Hilbert-Schmidt operators
${\displaystyle \ell ^{p}}$= Space of -fold summable sequences, ${\displaystyle p}$ ${\displaystyle 1<p<\infty }$	${\displaystyle {\mathcal {S}}_{p}(H)}$= - shadow class ${\displaystyle p}$
${\displaystyle \ell ^{\infty }}$ = Space of limited consequences	${\displaystyle B(H)}$ = Algebra of all bounded operators
${\displaystyle c_{00}\subset \ell ^{1}\subset c_{0}\subset \ell ^{\infty }}$	${\displaystyle F(H)\subset N(H)\subset K(H)\subset B(H)}$
${\displaystyle c_{00}}$lies close to the supreme norm . ${\displaystyle c_{0}}$ ${\displaystyle \|\cdot \|_{\infty }}$	${\displaystyle F(H)}$lies close in with respect to the operator norm. ${\displaystyle K(H)}$
${\displaystyle c_{00}}$lies close to the norm . ${\displaystyle \ell ^{1}}$ ${\displaystyle \|\cdot \|_{1}}$	${\displaystyle F(H)}$is close to the nuclear norm. ${\displaystyle N(H)}$
${\displaystyle \ell ^{1}}$lies close to the supreme norm. ${\displaystyle c_{0}}$	${\displaystyle N(H)}$lies close in with respect to the operator norm. ${\displaystyle K(H)}$
${\displaystyle c_{00}}$is an ideal in , and in . ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{\infty }}$	${\displaystyle F(H)}$is an ideal in , and in . ${\displaystyle K(H)}$ ${\displaystyle N(H)}$ ${\displaystyle B(H)}$
${\displaystyle \ell ^{1}}$is an ideal in and in . ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{\infty }}$	${\displaystyle N(H)}$is an ideal in and in . ${\displaystyle K(H)}$ ${\displaystyle B(H)}$
${\displaystyle c_{0}}$is an ideal in . ${\displaystyle \ell ^{\infty }}$	${\displaystyle K(H)}$is an ideal in . ${\displaystyle B(H)}$
${\displaystyle (a_{n})_{n}\mapsto \sum _{n=1}^{\infty }a_{n}}$is a continuous linear functional . ${\displaystyle \ell ^{1}}$	The trace is a continuous linear functional . ${\displaystyle N(H)}$
${\displaystyle (c_{0},\|\cdot \|_{\infty })'\cong (\ell ^{1},\|\cdot \|_{1})}$.	${\displaystyle (K(H),\|\cdot \|)'\cong (N(H),\|\cdot \|_{1})}$.
${\displaystyle (\ell ^{1},\|\cdot \|_{1})'\cong (\ell ^{\infty },\|\cdot \|_{\infty })}$.	${\displaystyle (N(H),\|\cdot \|_{1})'\cong (B(H),\|\cdot \|)}$.
A sequence out is out if and only if it is the product of two sequences. ${\displaystyle \ell ^{\infty }}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{2}}$	A continuous linear operator is nuclear if and only if it is the product of two Hilbert-Schmidt operators.

Nuclear operators on Banach spaces

The study of nuclear operators on Banach spaces began in 1951 with a paper by AF Ruston . Because of the missing orthonormal bases here, the relationships are not as simple as in the Hilbert space case, and clearly different methods are required.

While in the Hilbert space case the eigenvalue sequence of a nuclear operator can be absolutely summed according to the above, one can only prove the following weaker statement in the Banach space case:

If a Banach space and is the eigenvalue sequence of a nuclear operator , then and . ${\displaystyle E}$ ${\displaystyle (\lambda _{n})_{n}}$ ${\displaystyle A\in N(E)}$ ${\displaystyle (\lambda _{n})_{n}\in \ell ^{2}}$ ${\displaystyle \|(\lambda _{n})_{n}\|_{2}\leq \|A\|_{1}}$

This result cannot be improved; RJ Kaiser and James Ronald Retherford have specified a nuclear operator with this sequence of eigenvalues for the given sequence . According to a theorem of Johnson, König, Maurray and Retherford, a Banach space is isomorphic to a Hilbert space if and only if the eigenvalue sequence of every nuclear operator is off. ${\displaystyle \ell ^{2}}$ ${\displaystyle N(\ell ^{1}\oplus \ell ^{\infty })}$ ${\displaystyle \ell ^{1}}$

The trace of a nuclear operator cannot be defined for all Banach spaces. Given a nuclear representation of an operator from , the Hilbert space case suggests the definition . This number turns out to be well-defined, that is, to be independent of the chosen nuclear representation, if the Banach space has the approximation property . ${\displaystyle \textstyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)x_{n}}$ ${\displaystyle A\in N(E)}$ ${\displaystyle \textstyle \mathrm {Sp} (A)=\sum _{n=1}^{\infty }f_{n}(x_{n})}$

The duality present in the Hilbert space case generalizes to Banach spaces with approximation properties as follows . Each defines a continuous, linear functional on , wherein when a nuclear representation of is. The approximation property ensures the well-definedness, i.e. H. independence from the choice of nuclear representation. It can be shown that there is an isometric isomorphism if one adds the nuclear norm and the operator norm. With that in mind is . If, therefore, it is also reflexive , one has, as in the Hilbert space case. ${\displaystyle E}$ ${\displaystyle T\in B(E,E'')}$ ${\displaystyle \psi _{T}}$ ${\displaystyle N(E)}$ ${\displaystyle \textstyle \psi _{T}(A)=\sum _{n=1}^{\infty }(T(x_{n}))(f_{n})}$ ${\displaystyle \textstyle A(x)=\sum _{n=1}^{\infty }f_{n}(x)x_{n}}$ ${\displaystyle A}$ ${\displaystyle B(E,E'')\rightarrow N(E)\,',T\mapsto \psi _{T}}$ ${\displaystyle N(E)}$ ${\displaystyle B(E,E'')}$ ${\displaystyle N(E)'\cong B(E,E'')}$ ${\displaystyle E}$ ${\displaystyle N(E)'\cong B(E)}$

Nuclear operators on locally convex spaces

In 1951, Alexander Grothendieck began investigating nuclear operators between locally convex spaces. Since there is no norm available on locally convex spaces, the definition must be formulated as follows: A linear operator is called nuclear if it is a representation of the kind ${\displaystyle A:E\rightarrow F}$

${\displaystyle A(x)=\sum _{n=1}^{\infty }\lambda _{n}f_{n}(x)y_{n}}$

there, where

• ${\displaystyle (\lambda _{n})_{n}\in \ell ^{1}}$,
• ${\displaystyle (f_{n})_{n}}$is an equally continuous sequence in the strong dual space (i.e. there is a continuous semi-norm on with for all ), ${\displaystyle E\,'}$ ${\displaystyle p}$ ${\displaystyle E}$ ${\displaystyle |f_{n}(x)|\leq p(x)}$ ${\displaystyle x\in E}$
• ${\displaystyle (y_{n})_{n}}$is a limited sequence in . ${\displaystyle F}$

Since the required equality in the Banach space case equals the boundedness, the definition given here in the Banach space case leads to the same concept of the nuclear operator as defined above.

The ideal property is generalized to locally convex spaces: If nuclear and are and continuous, linear operators between locally convex spaces, then nuclear is also . Nuclear operators are continuous and, if complete , even compact. One can show that for every nuclear operator there is a further nuclear operator between normalized spaces and continuous linear operators with . With this one can trace the study of the nuclear operators between locally convex spaces back to the normalized case. ${\displaystyle A:E\rightarrow F}$ ${\displaystyle B:G\rightarrow E}$ ${\displaystyle C:F\rightarrow H}$ ${\displaystyle CAB:G\rightarrow H}$ ${\displaystyle A:E\rightarrow F}$ ${\displaystyle F}$ ${\displaystyle A:E\rightarrow F}$ ${\displaystyle {\tilde {A}}:G\rightarrow H}$ ${\displaystyle B:E\rightarrow G,C:H\rightarrow F}$ ${\displaystyle A=C{\tilde {A}}B}$

In the locally convex theory, the nuclear operators play an important role in connection with nuclear spaces .

Application in statistical physics

The physical field of statistical physics is based on the central assumption that the trace exists with the exponential function of the so-called Hamilton operator (energy operator) for the temperature- weighted measurand ( observable ) of quantum statistics , although the Hamilton operator itself does not belong to the trace class and usually the same applies to the (only self-adjoint!) operator . For the thermal expected value of the measured variable under consideration, the relationship still applies due to this assumption ${\displaystyle {\mathcal {H}}}$ ${\displaystyle T}$ ${\displaystyle {\hat {A}}}$ ${\displaystyle {\hat {A}}}$ ${\displaystyle \langle {\hat {A}}\rangle _{T}}$

${\displaystyle \langle {\hat {A}}\rangle _{T}=\mathrm {Sp} \{e^{-{\frac {\mathcal {H}}{T}}}\,{\hat {A}}\}\,.}$

In other words, the terms in brackets deal with i. W. with nuclear spaces and the operators or measured variables defined therein.

literature

• R. Schatten , J. v. Neumann : The Cross Space of Linear Transformations II . In: Ann. of Math. , 47, 1946, pp. 608-630
• AF Ruston: On the Fredholm theory of integral equations for operators belonging to the trace class of a general Banach space . In: Proc. London Math. Soc. , 2, 53, 1951, pp. 109-124
• A. Grothendieck: Sur une notion de produit tensoriel topologique d'espaces vectoriels topologiques, et une classe remarquable d'espaces vectoriels liée à cette notion . In: CR Acad. Sci. Paris , vol. 233, 1951, pp. 1556-1558
• A. Pietsch: Nuclear locally convex spaces . Akademie-Verlag, (1965)
• A. Pietsch: Eigenvalues ​​and s-Numbers . In: Cambridge Studies in Advanced Mathematics , 1987
• RJ Kaiser, JR Retherford: Presigning eigenvalues ​​and zeros of nuclear operators . In: Studia Math. , 81, 1985, pp. 127-133
• K. Floret, J. Wloka: Introduction to the theory of locally convex spaces . In: Lecture Notes in Mathematics , 56, p. 1968
• HH Schaefer: Topological Vector Spaces . Springer, 1971
• R. Meise, D. Vogt: Introduction to functional analysis . Vieweg, 1992
• Dirk Werner : Functional Analysis . 2nd Edition. Springer, Berlin / Heidelberg / New York 1997, ISBN 3-540-61904-6