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.