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

<math>\sum_{x\in \Omega}<Ax,x></math>

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

<math>\operatorname{tr}(aA+bB)=a\,\operatorname{tr}(A)+b\,\operatorname{tr}(B).</math>

The bilinear map <A,B>=tr(AB^*) is an inner product on the trace class, where the induced norm is called the trace norm.