Trace class
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 Ω of H; the sum
is finite. In this case, the sum is called the trace of A, denoted by tr(A) and is independent of the choice of the orthonormal bases.
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 trace class, meaning
The bilinear map <A,B>=tr(AB*) is an inner product on the trace class, where the induced norm is called the trace norm.
The set  of trace class operators on H is a two-sided ideal of 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 on by φ(A)=tr(AT). This correspondence between elements φ of the dual space of B(H) and trace-class operators is an isometric isomorphism. It follows that is the dual space of B(H). This is used to defined the weak-* topology on B(H).
Referenced By
List of functional analysis topics | List of mathematical topics (S-U)
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