Operator Trace

The operator trace of a trace-class operator on a complex Hilbert space is

where is any Hilbert basis of and denotes the inner product, taken to be linear in its first argument. The sum is absolutely convergent and independent of the chosen Hilbert basis (Bell 2016, Theorem 19).

For a positive operator , the same sum defines its operator trace in . It is finite iff is a trace-class operator (Anantharaman and Popa n.d., pp. 116, 124). In finite dimension, the operator trace agrees with the matrix trace of the representing matrix.