Trace of an Operator

Let $A:H\to H$ be a trace-class operator on a complex Hilbert space, using the inner product convention linear in its first argument. For any orthonormal basis $(e_j)_{j\in J}$ , the trace is

\operatorname{Tr}(A)=\sum_{j\in J}\langle A e_j,e_j\rangle.

The sum is absolutely convergent and independent of the basis. For an arbitrary index set, it is the limit over finite subsets; at most countably many summands are nonzero. In finite dimension every linear operator is trace-class, and this formula equals the matrix trace.

Equivalent descriptions (finite dimension)

If $A$ is represented by a matrix $(A_{ij})$ in any basis, then $\operatorname{Tr}(A)=\sum_i A_{ii}$ .
$\operatorname{Tr}(A)$ equals the sum of eigenvalues of $A$ , counted with algebraic multiplicity.

Properties

For trace-class operators $A,B$ on $H$ and scalars $\alpha,\beta$ :

Linearity: $\operatorname{Tr}(\alpha A+\beta B)=\alpha\operatorname{Tr}(A)+\beta\operatorname{Tr}(B)$ .
Cyclic property: $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$ .
Unitary invariance: if $U$ is unitary, then $\operatorname{Tr}(U^\ast A U)=\operatorname{Tr}(A)$ .
Positivity: if $A$ is positive semidefinite, then $\operatorname{Tr}(A)\ge 0$ .

The cyclic identity also holds when one factor is trace-class and the other is bounded; both products are then trace-class.

Quantum expectation values

For a density operator $\rho$ and a bounded self-adjoint observable $A$ ,

\mathbb E_\rho[A]=\operatorname{Tr}(\rho A).

For an unbounded observable, additional domain and integrability conditions are needed.

Positive operators outside trace class

For a bounded positive operator $A$ , the same sum of nonnegative diagonal entries defines an extended trace in $[0,\infty]$ , independent of the basis. Finiteness is equivalent to trace-class membership. An arbitrary bounded operator need not have a trace.

References

Dan-Virgil Voiculescu, Math 209: Von Neumann Algebras, notes by Leonard Tomczak, UC Berkeley, Spring 2024. Lecture notes, §4, pp. 7–8, Proposition 4.1 and the singular-value expansion.