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