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.
If is in the trace class, we define the trace of by
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.
Given a bounded linear operator , each of the following statements is equivalent to being in the trace class:
- For some orthonormal basis of H, the sum of positive terms is finite.
- For every orthonormal basis of H, the sum of positive terms is finite.
- T is a compact operator and where are the eigenvalues of (also known as the singular values of T) with each eigenvalue repeated as often as its multiplicity.
- There exist two orthogonal sequences and in and a sequence in such that for all  Here, the infinite sum means that the sequence of partial sums converges to in H.
- T is a nuclear operator.
- T is equal to the composition of two Hilbert-Schmidt operators.
- is a Hilbert-Schmidt operator.
- T is an integral operator.
- There exist weakly closed and equicontinuous (and thus weakly compact) subsets and of and respectively, and some positive Radon measure on of total mass such that for all and :
We define the trace-norm of a trace class operator T to be the value
If T is trace class then
Every bounded linear operator that has a finite-dimensional range (i.e. operators of finite-rank) is trace class; furthermore, the space of all finite-rank operators is a dense subspace of (when endowed with the norm). The composition of two Hilbert-Schmidt operators is a trace class operator.
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), 
- 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,
- is a positive linear functional such that if is a trace class operator satisfying then 
- If is trace-class then so is and 
- 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 
- If and are two orthonormal bases of H and if T is trace class then 
- If A is trace-class, then one can define the Fredholm determinant of :
- If is trace class then for any orthonormal basis of the sum of positive terms is finite.
- If for some Hilbert-Schmidt operators and then for any normal vector holds.
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
Relationship between some 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 (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
The trace-class operators are given the trace norm The norm corresponding to the Hilbert–Schmidt inner product is
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
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
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 continuous linear 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
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