Trace class
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In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms
- <math>\sum_{k} \langle (A^*A)^{1/2} \, e_k, e_k \rangle<math>
is finite. In this case, the sum
- <math>\sum_{k} \langle A e_k, e_k \rangle<math>
is absolutely convergent and is independent of the choice of the orthonormal basis. This value is called the trace of A, denoted by Tr(A).
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
- <math>\sum_{k} \langle A e_k, e_k \rangle. <math>
If A is a non-negative self-adjoint, A is trace class iff Tr(A) < ∞. An operator A is trace class iff its positive part A+ and negative part A- are both trace class.
When H is finite-dimensional, then the trace of A is just the trace of a matrix and the last property stated above is roughly saying that trace is invariant under similarity.
The trace is a linear functional over the space of trace class operators, meaning
- <math>\operatorname{Tr}(aA+bB)=a\,\operatorname{Tr}(A)+b\,\operatorname{Tr}(B).<math>
The bilinear map
- <math> \langle A, B \rangle = \operatorname{Tr}(A^* B) <math>
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 can also be considered as a class of operators, the Hilbert-Schmidt operators.
For infinite dimensional spaces, the class of Hilbert-Schmidt operators is strictly larger than that of trace class operators. The heuristic is that Hilbert-Schmidt is to trace class as l2(N) is to l1(N).
The set <math>C_1<math> of trace class operators on H is a two-sided ideal in B(H), the set of all bounded linear operators on H. So given any operator T in B(H), we may define a continuous linear functional φT on <math>C_1<math> by φT(A)=Tr(AT). This correspondence between elements φT of the dual space of <math>C_1<math> and bounded linear operators is an isometric isomorphism. It follows that B(H) is the dual space of <math>C_1<math>. This can be used to defined the weak-* topology on B(H).
References
- Dixmier, J. (1969). Les Algebres d'Operateurs dans l'Espace Hilbertien. Gauthier-Villars.