Trace class

In mathematics, a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and reserve "nuclear operator" for usage in more general topological vector spaces (such as Banach spaces).

It may be shown that a bounded linear operator on a Hilbert space is trace class if and only if its absolute value is trace class.[1]

When H is finite-dimensional, every operator is trace class and this definition of trace of A coincides with the definition of the trace of a matrix.

By extension, if A is a non-negative self-adjoint operator, we can also define the trace of A as an extended real number by the possibly divergent sum

Note that the series in the left converges absolutely due to Weyl's inequality

It is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms.