In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators.
Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces).
Note that the trace operator studied in partial differential equations is an unrelated concept.
The trace-norm of a trace class operator T is defined as
One can show that the trace-norm is a norm on the space of all trace class operators and that , with the trace-norm, becomes a Banach space.
When H is finite-dimensional, every operator is trace class and this definition of trace of T coincides with the definition of the trace of a matrix. If H is complex, then is always self-adjoint (i.e. ) though the converse is not necessarily true.[4]
Equivalent formulationsedit
Given a bounded linear operator , each of the following statements is equivalent to being in the trace class:
Every finite-rank operator is a trace-class operator. Furthermore, the space of all finite-rank operators is a dense subspace of (when endowed with the trace norm).[8]
Given any define the operator by Then is a continuous linear operator of rank 1 and is thus trace class; moreover, for any bounded linear operator A on H (and into H), [8]
Propertiesedit
If is a non-negative self-adjoint operator, then is trace-class if and only if Therefore, a self-adjoint operator is trace-class if and only if its positive part and negative part are both trace-class. (The positive and negative parts of a self-adjoint operator are obtained by the continuous functional calculus.)
The trace is a linear functional over the space of trace-class operators, that is,
The bilinear map
is an inner product on the trace class; the corresponding norm is called the Hilbert–Schmidt norm. The completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.
is a positive linear functional such that if is a trace class operator satisfying then [10]
If is bounded, and is trace-class, then and are also trace-class (i.e. the space of trace-class operators on H is an ideal in the algebra of bounded linear operators on H), and[10][12]
Let be a trace-class operator in a separable Hilbert space and let be the eigenvalues of Let us assume that are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of is then is repeated times in the list ). Lidskii's theorem (named after Victor Borisovich Lidskii) states that
Note that the series on the right converges absolutely due to Weyl's inequality
between the eigenvalues and the singular values of the compact operator [13]
Relationship between common classes of operatorsedit
One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces, with trace-class operators as the noncommutative analogue of the sequence space
Indeed, it is possible to apply the spectral theorem to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an sequence with respect to some choice of a pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of the compact operators that of (the sequences convergent to 0), Hilbert–Schmidt operators correspond to and finite-rank operators to (the sequences that have only finitely many non-zero terms). To some extent, the relationships between these classes of operators are similar to the relationships between their commutative counterparts.
Recall that every compact operator on a Hilbert space takes the following canonical form: there exist orthonormal bases and and a sequence of non-negative numbers with such that
Making the above heuristic comments more precise, we have that is trace-class iff the series is convergent, is Hilbert–Schmidt iff is convergent, and is finite-rank iff the sequence has only finitely many nonzero terms. This allows to relate these classes of operators. The following inclusions hold and are all proper when is infinite-dimensional:
The trace-class operators are given the trace norm The norm corresponding to the Hilbert–Schmidt inner product is
Also, the usual operator norm is By classical inequalities regarding sequences,
for appropriate
It is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms.
Trace class as the dual of compact operatorsedit
The dual space of is Similarly, we have that the dual of compact operators, denoted by is the trace-class operators, denoted by The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let we identify with the operator defined by
where is the rank-one operator given by
This identification works because the finite-rank operators are norm-dense in In the event that is a positive operator, for any orthonormal basis one has
where is the identity operator:
But this means that is trace-class. An appeal to polar decomposition extend this to the general case, where need not be positive.
A limiting argument using finite-rank operators shows that Thus is isometrically isomorphic to
As the predual of bounded operatorsedit
Recall that the dual of is In the present context, the dual of trace-class operators is the bounded operators More precisely, the set is a two-sided ideal in So given any operator we may define a continuouslinear functional on by This correspondence between bounded linear operators and elements of the dual space of is an isometric isomorphism. It follows that is the dual space of This can be used to define the weak-* topology on
Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN978-1-4612-7155-0. OCLC 840278135.
Simon, Barry (2010). Szegő's theorem and its descendants: spectral theory for L² perturbations of orthogonal polynomials. Princeton University Press. ISBN978-0-691-14704-8.
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In mathematics specifically functional analysis a trace class operator is a linear operator for which a trace may be defined such that the trace is a finite number independent of the choice of basis used to compute the trace This trace of trace class operators generalizes the trace of matrices studied in linear algebra All trace class operators are compact operators In quantum mechanics mixed states are described by density matrices which are certain trace class operators Trace class operators are essentially the same as nuclear operators though many authors reserve the term trace class operator for the special case of nuclear operators on Hilbert spaces and use the term nuclear operator in more general topological vector spaces such as Banach spaces Note that the trace operator studied in partial differential equations is an unrelated concept Contents 1 Definition 2 Equivalent formulations 3 Examples 3 1 Spectral theorem 3 2 Mercer s theorem 3 3 Finite rank operators 4 Properties 4 1 Lidskii s theorem 4 2 Relationship between common classes of operators 4 3 Trace class as the dual of compact operators 4 4 As the predual of bounded operators 5 See also 6 References 7 BibliographyDefinition editLet H displaystyle H nbsp be a separable Hilbert space e k k 1 displaystyle left e k right k 1 infty nbsp an orthonormal basis and T H H displaystyle T H to H nbsp a positive bounded linear operator on H displaystyle H nbsp The trace of T displaystyle T nbsp is denoted by Tr T displaystyle operatorname Tr T nbsp and defined as 1 2 Tr T k 1 T e k e k displaystyle operatorname Tr T sum k 1 infty left langle Te k e k right rangle nbsp independent of the choice of orthonormal basis The operator T displaystyle T nbsp is called trace class if and only if Tr T lt displaystyle operatorname Tr T lt infty nbsp where T T T displaystyle T sqrt T T nbsp denotes the positive semidefinite Hermitian square root 3 The trace norm of a trace class operator T is defined as T 1 Tr T displaystyle T 1 operatorname Tr T nbsp One can show that the trace norm is a norm on the space of all trace class operators B 1 H displaystyle B 1 H nbsp and that B 1 H displaystyle B 1 H nbsp with the trace norm becomes a Banach space When H is finite dimensional every operator is trace class and this definition of trace of T coincides with the definition of the trace of a matrix If H is complex then T displaystyle T nbsp is always self adjoint i e T T T displaystyle T T T nbsp though the converse is not necessarily true 4 Equivalent formulations editGiven a bounded linear operator T H H displaystyle T H to H nbsp each of the following statements is equivalent to T displaystyle T nbsp being in the trace class Tr T k T e k e k textstyle operatorname Tr T sum k left langle T e k e k right rangle nbsp is finite for every orthonormal basis e k k displaystyle left e k right k nbsp of H 1 T is a nuclear operator 5 6 There exist two orthogonal sequences x i i 1 displaystyle left x i right i 1 infty nbsp and y i i 1 displaystyle left y i right i 1 infty nbsp in H displaystyle H nbsp and positive real numbers l i i 1 displaystyle left lambda i right i 1 infty nbsp in ℓ 1 displaystyle ell 1 nbsp such that i 1 l i lt textstyle sum i 1 infty lambda i lt infty nbsp andx T x i 1 l i x x i y i x H displaystyle x mapsto T x sum i 1 infty lambda i left langle x x i right rangle y i quad forall x in H nbsp dd where l i i 1 displaystyle left lambda i right i 1 infty nbsp are the singular values of T or equivalently the eigenvalues of T displaystyle T nbsp with each value repeated as often as its multiplicity 7 dd T is a compact operator with Tr T lt displaystyle operatorname Tr T lt infty nbsp If T is trace class then 8 T 1 sup Tr C T C 1 and C H H is a compact operator displaystyle T 1 sup left operatorname Tr CT C leq 1 text and C H to H text is a compact operator right nbsp dd dd T is an integral operator 9 T is equal to the composition of two Hilbert Schmidt operators 10 T textstyle sqrt T nbsp is a Hilbert Schmidt operator 10 Examples editSpectral theorem edit Let T displaystyle T nbsp be a bounded self adjoint operator on a Hilbert space Then T 2 displaystyle T 2 nbsp is trace class if and only if T displaystyle T nbsp has a pure point spectrum with eigenvalues l i T i 1 displaystyle left lambda i T right i 1 infty nbsp such that 11 Tr T 2 i 1 l i T 2 lt displaystyle operatorname Tr T 2 sum i 1 infty lambda i T 2 lt infty nbsp Mercer s theorem edit Mercer s theorem provides another example of a trace class operator That is suppose K displaystyle K nbsp is a continuous symmetric positive definite kernel on L 2 a b displaystyle L 2 a b nbsp defined as K s t j 1 l j e j s e j t displaystyle K s t sum j 1 infty lambda j e j s e j t nbsp then the associated Hilbert Schmidt integral operator T K displaystyle T K nbsp is trace class i e Tr T K a b K t t d t i l i displaystyle operatorname Tr T K int a b K t t dt sum i lambda i nbsp Finite rank operators edit Every finite rank operator is a trace class operator Furthermore the space of all finite rank operators is a dense subspace of B 1 H displaystyle B 1 H nbsp when endowed with the trace norm 8 Given any x y H displaystyle x y in H nbsp define the operator x y H H displaystyle x otimes y H to H nbsp by x y z z y x displaystyle x otimes y z langle z y rangle x nbsp Then x y displaystyle x otimes y nbsp is a continuous linear operator of rank 1 and is thus trace class moreover for any bounded linear operator A on H and into H Tr A x y A x y displaystyle operatorname Tr A x otimes y langle Ax y rangle nbsp 8 Properties editIf A H H displaystyle A H to H nbsp is a non negative self adjoint operator then A displaystyle A nbsp is trace class if and only if Tr A lt displaystyle operatorname Tr A lt infty nbsp Therefore a self adjoint operator A displaystyle A nbsp is trace class if and only if its positive part A displaystyle A nbsp and negative part A displaystyle A nbsp are both trace class The positive and negative parts of a self adjoint operator are obtained by the continuous functional calculus The trace is a linear functional over the space of trace class operators that is Tr a A b B a Tr A b Tr B displaystyle operatorname Tr aA bB a operatorname Tr A b operatorname Tr B nbsp The bilinear map A B Tr A B displaystyle langle A B rangle operatorname Tr A B nbsp is an inner product on the trace class the corresponding norm is called the Hilbert Schmidt norm The completion of the trace class operators in the Hilbert Schmidt norm are called the Hilbert Schmidt operators Tr B 1 H C displaystyle operatorname Tr B 1 H to mathbb C nbsp is a positive linear functional such that if T displaystyle T nbsp is a trace class operator satisfying T 0 and Tr T 0 displaystyle T geq 0 text and operatorname Tr T 0 nbsp then T 0 displaystyle T 0 nbsp 10 If T H H displaystyle T H to H nbsp is trace class then so is T displaystyle T nbsp and T 1 T 1 displaystyle T 1 left T right 1 nbsp 10 If A H H displaystyle A H to H nbsp is bounded and T H H displaystyle T H to H nbsp is trace class then A T displaystyle AT nbsp and T A displaystyle TA nbsp are also trace class i e the space of trace class operators on H is an ideal in the algebra of bounded linear operators on H and 10 12 A T 1 Tr A T A T 1 T A 1 Tr T A A T 1 displaystyle AT 1 operatorname Tr AT leq A T 1 quad TA 1 operatorname Tr TA leq A T 1 nbsp Furthermore under the same hypothesis 10 Tr A T Tr T A displaystyle operatorname Tr AT operatorname Tr TA nbsp and Tr A T A T displaystyle operatorname Tr AT leq A T nbsp The last assertion also holds under the weaker hypothesis that A and T are Hilbert Schmidt If e k k displaystyle left e k right k nbsp and f k k displaystyle left f k right k nbsp are two orthonormal bases of H and if T is trace class then k T e k f k T 1 textstyle sum k left left langle Te k f k right rangle right leq T 1 nbsp 8 If A is trace class then one can define the Fredholm determinant of I A displaystyle I A nbsp det I A n 1 1 l n A displaystyle det I A prod n geq 1 1 lambda n A nbsp where l n A n displaystyle lambda n A n nbsp is the spectrum of A displaystyle A nbsp The trace class condition on A displaystyle A nbsp guarantees that the infinite product is finite indeed det I A e A 1 displaystyle det I A leq e A 1 nbsp It also implies that det I A 0 displaystyle det I A neq 0 nbsp if and only if I A displaystyle I A nbsp is invertible If A H H displaystyle A H to H nbsp is trace class then for any orthonormal basis e k k displaystyle left e k right k nbsp of H displaystyle H nbsp the sum of positive terms k A e k e k textstyle sum k left left langle A e k e k right rangle right nbsp is finite 10 If A B C displaystyle A B C nbsp for some Hilbert Schmidt operators B displaystyle B nbsp and C displaystyle C nbsp then for any normal vector e H displaystyle e in H nbsp A e e 1 2 B e 2 C e 2 textstyle langle Ae e rangle frac 1 2 left Be 2 Ce 2 right nbsp holds 10 Lidskii s theorem edit Let A displaystyle A nbsp be a trace class operator in a separable Hilbert space H displaystyle H nbsp and let l n A n 1 N displaystyle lambda n A n 1 N leq infty nbsp be the eigenvalues of A displaystyle A nbsp Let us assume that l n A displaystyle lambda n A nbsp are enumerated with algebraic multiplicities taken into account that is if the algebraic multiplicity of l displaystyle lambda nbsp is k displaystyle k nbsp then l displaystyle lambda nbsp is repeated k displaystyle k nbsp times in the list l 1 A l 2 A displaystyle lambda 1 A lambda 2 A dots nbsp Lidskii s theorem named after Victor Borisovich Lidskii states thatTr A n 1 N l n A displaystyle operatorname Tr A sum n 1 N lambda n A nbsp Note that the series on the right converges absolutely due to Weyl s inequality n 1 N l n A m 1 M s m A displaystyle sum n 1 N left lambda n A right leq sum m 1 M s m A nbsp between the eigenvalues l n A n 1 N displaystyle lambda n A n 1 N nbsp and the singular values s m A m 1 M displaystyle s m A m 1 M nbsp of the compact operator A displaystyle A nbsp 13 Relationship between common classes of operators edit One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces with trace class operators as the noncommutative analogue of the sequence space ℓ 1 N displaystyle ell 1 mathbb N nbsp Indeed it is possible to apply the spectral theorem to show that every normal trace class operator on a separable Hilbert space can be realized in a certain way as an ℓ 1 displaystyle ell 1 nbsp sequence with respect to some choice of a pair of Hilbert bases In the same vein the bounded operators are noncommutative versions of ℓ N displaystyle ell infty mathbb N nbsp the compact operators that of c 0 displaystyle c 0 nbsp the sequences convergent to 0 Hilbert Schmidt operators correspond to ℓ 2 N displaystyle ell 2 mathbb N nbsp and finite rank operators to c 00 displaystyle c 00 nbsp the sequences that have only finitely many non zero terms To some extent the relationships between these classes of operators are similar to the relationships between their commutative counterparts Recall that every compact operator T displaystyle T nbsp on a Hilbert space takes the following canonical form there exist orthonormal bases u i i displaystyle left u i right i nbsp and v i i displaystyle left v i right i nbsp and a sequence a i i displaystyle left alpha i right i nbsp of non negative numbers with a i 0 displaystyle alpha i to 0 nbsp such thatT x i a i x v i u i for all x H displaystyle Tx sum i alpha i langle x v i rangle u i quad text for all x in H nbsp Making the above heuristic comments more precise we have that T displaystyle T nbsp is trace class iff the series i a i textstyle sum i alpha i nbsp is convergent T displaystyle T nbsp is Hilbert Schmidt iff i a i 2 textstyle sum i alpha i 2 nbsp is convergent and T displaystyle T nbsp is finite rank iff the sequence a i i displaystyle left alpha i right i nbsp has only finitely many nonzero terms This allows to relate these classes of operators The following inclusions hold and are all proper when H displaystyle H nbsp is infinite dimensional finite rank trace class Hilbert Schmidt compact displaystyle text finite rank subseteq text trace class subseteq text Hilbert Schmidt subseteq text compact nbsp The trace class operators are given the trace norm T 1 Tr T T 1 2 i a i textstyle T 1 operatorname Tr left left T T right 1 2 right sum i alpha i nbsp The norm corresponding to the Hilbert Schmidt inner product is T 2 Tr T T 1 2 i a i 2 1 2 displaystyle T 2 left operatorname Tr left T T right right 1 2 left sum i alpha i 2 right 1 2 nbsp Also the usual operator norm is T sup i a i textstyle T sup i left alpha i right nbsp By classical inequalities regarding sequences T T 2 T 1 displaystyle T leq T 2 leq T 1 nbsp for appropriate T displaystyle T nbsp It is also clear that finite rank operators are dense in both trace class and Hilbert Schmidt in their respective norms Trace class as the dual of compact operators edit The dual space of c 0 displaystyle c 0 nbsp is ℓ 1 N displaystyle ell 1 mathbb N nbsp Similarly we have that the dual of compact operators denoted by K H displaystyle K H nbsp is the trace class operators denoted by B 1 displaystyle B 1 nbsp The argument which we now sketch is reminiscent of that for the corresponding sequence spaces Let f K H displaystyle f in K H nbsp we identify f displaystyle f nbsp with the operator T f displaystyle T f nbsp defined by T f x y f S x y displaystyle langle T f x y rangle f left S x y right nbsp where S x y displaystyle S x y nbsp is the rank one operator given by S x y h h y x displaystyle S x y h langle h y rangle x nbsp This identification works because the finite rank operators are norm dense in K H displaystyle K H nbsp In the event that T f displaystyle T f nbsp is a positive operator for any orthonormal basis u i displaystyle u i nbsp one has i T f u i u i f I f displaystyle sum i langle T f u i u i rangle f I leq f nbsp where I displaystyle I nbsp is the identity operator I i u i u i displaystyle I sum i langle cdot u i rangle u i nbsp But this means that T f displaystyle T f nbsp is trace class An appeal to polar decomposition extend this to the general case where T f displaystyle T f nbsp need not be positive A limiting argument using finite rank operators shows that T f 1 f displaystyle T f 1 f nbsp Thus K H displaystyle K H nbsp is isometrically isomorphic to C 1 displaystyle C 1 nbsp As the predual of bounded operators edit Recall that the dual of ℓ 1 N displaystyle ell 1 mathbb N nbsp is ℓ N displaystyle ell infty mathbb N nbsp In the present context the dual of trace class operators B 1 displaystyle B 1 nbsp is the bounded operators B H displaystyle B H nbsp More precisely the set B 1 displaystyle B 1 nbsp is a two sided ideal in B H displaystyle B H nbsp So given any operator T B H displaystyle T in B H nbsp we may define a continuous linear functional f T displaystyle varphi T nbsp on B 1 displaystyle B 1 nbsp by f T A Tr A T displaystyle varphi T A operatorname Tr AT nbsp This correspondence between bounded linear operators and elements f T displaystyle varphi T nbsp of the dual space of B 1 displaystyle B 1 nbsp is an isometric isomorphism It follows that B H displaystyle B H nbsp is the dual space of C 1 displaystyle C 1 nbsp This can be used to define the weak topology on B H displaystyle B H nbsp See also editNuclear operator Nuclear operators between Banach spaces Trace operatorReferences edit a b Conway 2000 p 86 Reed amp Simon 1980 p 206 Reed amp Simon 1980 p 196 Reed amp Simon 1980 p 195 Treves 2006 p 494 Conway 2000 p 89 Reed amp Simon 1980 pp 203 204 209 a b c d Conway 1990 p 268 Treves 2006 pp 502 508 a b c d e f g h Conway 1990 p 267 Simon 2010 p 21 Reed amp Simon 1980 p 218 Simon B 2005 Trace ideals and their applications Second Edition American Mathematical Society Bibliography editConway John B 2000 A Course in Operator Theory Providence R I American Mathematical Soc ISBN 978 0 8218 2065 0 Conway John B 1990 A course in functional analysis New York Springer Verlag ISBN 978 0 387 97245 9 OCLC 21195908 Dixmier J 1969 Les Algebres d Operateurs dans l Espace Hilbertien Gauthier Villars Reed M Simon B 1980 Methods of Modern Mathematical Physics Vol 1 Functional analysis Academic Press ISBN 978 0 12 585050 6 Schaefer Helmut H 1999 Topological Vector Spaces GTM Vol 3 New York NY Springer New York Imprint Springer ISBN 978 1 4612 7155 0 OCLC 840278135 Simon Barry 2010 Szego s theorem and its descendants spectral theory for L perturbations of orthogonal polynomials Princeton University Press ISBN 978 0 691 14704 8 Treves Francois 2006 1967 Topological Vector Spaces Distributions and Kernels Mineola N Y Dover Publications ISBN 978 0 486 45352 1 OCLC 853623322 Retrieved from https en wikipedia org w index php title Trace class amp oldid 1204926802, wikipedia, wiki, book, books, library,