An operator on a Hilbert space
is compact if it can be written in the form[citation needed]
where and and are (not necessarily complete) orthonormal sets. Here is a set of real numbers, the set of singular values of the operator, obeying if
The bracket is the scalar product on the Hilbert space; the sum on the right hand side must converge in norm.
An operator that is compact as defined above is said to be nuclear or trace-class if
Properties
A nuclear operator on a Hilbert space has the important property that a trace operation may be defined. Given an orthonormal basis for the Hilbert space, the trace is defined as
Obviously, the sum converges absolutely, and it can be proven that the result is independent of the basis[citation needed]. It can be shown that this trace is identical to the sum of the eigenvalues of (counted with multiplicity).