Trace class

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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 (=trace class) operators for more general Banach spaces.

Contents 1 Definition 2 Properties 3 See also 4 References

Definition

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>

Properties

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).

See also

• Nuclear operator

References

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