Trace class
In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. 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 reserve "nuclear operator" for usage in more general Banach spaces.
Contents
Definition
A bounded linear operator A over a separable Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {e_{k}}_{k} of H, the sum of positive terms
is finite. In this case, the trace of A, which is given by the sum
is absolutely convergent and is independent of the choice of the orthonormal basis. When H is finite-dimensional, every operator is trace class and this definition of trace of A coincides with the definition of the trace of a matrix.
By extension, if A is a non-negative self-adjoint operator, we can also define the trace of A as an extended real number by the possibly divergent sum
Properties
1. | If A is a non-negative self-adjoint, A is trace class if and only if Tr(A) < ∞. Therefore a self adjoint operator A is trace class if and only if its positive part A^{+} and negative part A^{−} are both trace class. (The positive and negative parts of a self adjoint operator are obtained via the continuous functional calculus.) |
2. | The trace is a linear functional over the space of trace class operators, i.e.
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. |
3. | If is bounded and is trace class, and are also trace class and ^{[1]}
and furthermore, under the same hypothesis, The last assertion also holds under the weaker hypothesis that and are Hilbert Schmidt. |
4. | If is trace class, then one can define the Fredholm determinant of
where is the spectrum of . The trace class condition on guarantees that the infinite product is finite: indeed
It also implies that if and only if is invertible. |
Lidskii's theorem
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 (i.e. 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 in the left hand side converges absolutely due to Weyl's inequality
between the eigenvalues and the singular values of a compact operator . See e.g.^{[2]}
Relationship between some classes of operators
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).
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} 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), the compact operators that of c_{0} (the sequences convergent to 0), Hilbert–Schmidt operators correspond to l^{2}(N), 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 T on a Hilbert space takes the following canonical form
for some orthonormal bases {u_{i}} and {v_{i}}. Making the above heuristic comments more precise, we have that T is trace class if the series ∑_{i} α_{i} is convergent, T is Hilbert–Schmidt if ∑_{i} α_{i}^{2} is convergent, and T is finite rank if the sequence {α_{i}} has only finitely many nonzero terms.
The above description allows one to obtain easily some facts that relate these classes of operators. For example, the following inclusions hold and they are all proper when H is infinite dimensional: {finite rank} ⊂ {trace class} ⊂ {Hilbert–Schmidt} ⊂ {compact}.
The trace-class operators are given the trace norm ||T||_{1} = Tr [ (T*T)^{½} ] = ∑_{i} α_{i}. The norm corresponding to the Hilbert–Schmidt inner product is ||T||_{2} = (Tr T*T)^{½} = (∑_{i}α_{i}^{2})^{½}. Also, the usual operator norm is ||T|| = sup_{i}(α_{i}). By classical inequalities regarding sequences,
for appropriate T.
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
The dual space of c_{0} is l^{1}(N). Similarly, we have that the dual of compact operators, denoted by K(H)*, is the trace-class operators, denoted by C_{1}. The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let f ∈ K(H)*, we identify f with the operator T_{f} defined by
where S_{x,y} is the rank-one operator given by
This identification works because the finite-rank operators are norm-dense in K(H). In the event that T_{f} is a positive operator, for any orthonormal basis u_{i}, one has
where I is the identity operator
But this means T_{f} is trace-class. An appeal to polar decomposition extend this to the general case where T_{f} need not be positive.
A limiting argument via finite-rank operators shows that ||T_{f}||_{1} = ||f||. Thus K(H)* is isometrically isomorphic to C_{1}.
As the predual of bounded operators
Recall that the dual of l^{1}(N) is l^{∞}(N). In the present context, the dual of trace-class operators C_{1} is the bounded operators B(H). More precisely, the set C_{1} is a two-sided ideal in B(H). So given any operator T in B(H), we may define a continuous linear functional φ_{T} on by φ_{T}(A)=Tr(AT). This correspondence between bounded linear operators and elements φ_{T} of the dual space of is an isometric isomorphism. It follows that B(H) is the dual space of . This can be used to define the weak-* topology on B(H).
Notes
References
- Dixmier, J. (1969). Les Algebres d'Operateurs dans l'Espace Hilbertien. Gauthier-Villars.