In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace.
Note that the trace operator studied in partial differential equations is an unrelated concept.
and defined as[2][3] independent of the choice of orthonormal basis.
A (not necessarily positive) bounded linear operator
denotes the positive-semidefinite Hermitian square root.
[4] The trace-norm of a trace class operator T is defined as
One can show that the trace-norm is a norm on the space of all trace class operators
is finite-dimensional, every (positive) operator is trace class.
be a bounded self-adjoint operator on a Hilbert space.
such that[12] Mercer's theorem provides another example of a trace class operator.
Furthermore, the space of all finite-rank operators is a dense subspace of
be a trace-class operator in a separable Hilbert space
Note that the series on the right converges absolutely due to Weyl's inequality
[14] One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces, with trace-class operators as the noncommutative analogue of the sequence space
Indeed, it is possible to apply the spectral theorem to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an
sequence with respect to some choice of a pair of Hilbert bases.
In the same vein, the bounded operators are noncommutative versions of
(the sequences that have only finitely many non-zero terms).
on a Hilbert space takes the following canonical form: there exist orthonormal bases
Making the above heuristic comments more precise, we have that
{\displaystyle \{{\text{ finite rank }}\}\subseteq \{{\text{ trace class }}\}\subseteq \{{\text{ Hilbert--Schmidt }}\}\subseteq \{{\text{ compact }}\}.}
It is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms.
Similarly, we have that the dual of compact operators, denoted by
The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces.
This identification works because the finite-rank operators are norm-dense in
is a positive operator, for any orthonormal basis
An appeal to polar decomposition extend this to the general case, where
A limiting argument using finite-rank operators shows that
In the present context, the dual of trace-class operators
This correspondence between bounded linear operators and elements