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 {e_k}_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 φ_T on <math>C_1<math> by φ_T(A)=Tr(AT). This correspondence between elements φ_T 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

J. Dixmier, Les Algebres d'Operateurs dans l'Espace Hilbertien, Gauthier-Villars , 1969