  ## Encyclopedia > Trace class

Article Content

# 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
$\sum_{x\in \Omega}<Ax,x>$
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

$\operatorname{tr}(aA+bB)=a\,\operatorname{tr}(A)+b\,\operatorname{tr}(B).$
The bilinear map <A,B>=tr(AB*) is an inner product on the trace class, where the induced norm is called the trace norm.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Kings Park, New York ... there are 94.9 males. For every 100 females age 18 and over, there are 92.0 males. The median income for a household in the town is \$69,819, and the median income for a ...  