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

is finite. In this case, the sum

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

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

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 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 C₁ 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 C₁ by φ_T(A)=Tr(AT). This correspondence between elements φ_T of the dual space of C₁ and bounded linear operators is an isometric isomorphism. It follows that B(H) is the dual space of C₁. 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