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
, 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
) is an inner product
on the trace class, where the induced norm is called the trace norm