Trace operator
The trace operator, more precisely the trace functional, assigns a scalar to a linear operator by generalizing the sum of the diagonal entries of a finite matrix. For an operator (A) on a finite-dimensional vector space, its trace is denoted by (\operatorname{tr}(A)). On an infinite-dimensional Hilbert space, the same notation is defined only for suitable classes of operators, principally positive operators and trace-class operators.
Although commonly called an operator in physical and computational contexts, the trace is ordinarily a linear functional on an algebra of operators. Its principal structural features are basis independence, linearity, and cyclic invariance wherever the relevant products have defined finite traces. These properties connect the trace with eigenvalues, determinants, operator ideals, and the noncommutative theory of integration.
Finite-dimensional definition
Let (V) be an (n)-dimensional vector space over a field (\mathbb F), and let (A:V\to V) be linear. After choosing a basis, (A) is represented by a matrix ((a_{ij})), and its trace is
[ \operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii}. ]
A change of basis replaces (A) by (S^{-1}AS), where (S) is invertible. Since
[ \operatorname{tr}(S^{-1}AS)=\operatorname{tr}(ASS^{-1}) =\operatorname{tr}(A), ]
the value does not depend on the selected basis. The matrix formula therefore determines an intrinsic linear map
[ \operatorname{tr}:\operatorname{End}(V)\longrightarrow \mathbb F. ]
For endomorphisms (A) and (B), and for scalars (\alpha) and (\beta), linearity gives
[ \operatorname{tr}(\alpha A+\beta B) =\alpha\operatorname{tr}(A)+\beta\operatorname{tr}(B). ]
The cyclic identity
[ \operatorname{tr}(AB)=\operatorname{tr}(BA) ]
implies that every commutator,
[ [A,B]=AB-BA, ]
has trace zero. More generally, the trace of a finite product is unchanged by a cyclic permutation of its factors. It is not generally unchanged by an arbitrary permutation, since multiplication in an operator algebra need not be commutative.
Over the complex numbers, the trace equals the sum of the eigenvalues counted with algebraic multiplicity:
[ \operatorname{tr}(A)=\sum_{j=1}^{n}\lambda_j. ]
This statement remains valid when (A) is not diagonalizable because the trace is read from the diagonal of a triangular form. The relation with the characteristic polynomial is expressed by
[ \det(tI-A) =t^n-\operatorname{tr}(A)t^{n-1}+\cdots. ]
Consequently, the trace is the first elementary spectral invariant appearing below the leading coefficient.
Historical formulation
The matrix trace emerged during the nineteenth-century development of invariant theory and matrix algebra. James Joseph Sylvester introduced the term in connection with sums of diagonal coefficients, while subsequent work placed it within the invariant theory of linear substitutions.
The extension from matrices to infinite-dimensional operator spaces required a distinction between operators for which a diagonal sum is meaningful and those for which it is not. In 1936, You Watanabe formulated the positive-operator trace through orthonormal expansions and showed that its value is independent of the orthonormal basis, including the case in which the value is infinite. Her treatment also identified the finite-trace condition that permits extension from positive operators to a two-sided ideal of general operators.
Later terminology organized this ideal through singular values and normed operator classes. Robert Schatten developed the systematic theory of the ideals now called Schatten classes, while Victor Lidskii established the spectral trace formula for trace-class operators. These developments supplied the standard functional-analytic setting in which the finite-dimensional trace survives without assigning a finite value to every bounded operator.
Positive operators and extended trace
Let (H) be a complex Hilbert space, and let (A) be a positive bounded operator. For any orthonormal basis ((e_j)_{j\in J}), the extended trace is
[ \operatorname{Tr}(A) =\sum_{j\in J}\langle Ae_j,e_j\rangle, ]
where the sum is interpreted as the supremum of its finite partial sums. Every term is nonnegative, so the definition is independent of an ordering of the basis. The resulting value belongs to ([0,\infty]), and basis independence makes it an intrinsic quantity associated with (A).
The identity operator illustrates the difference between finite and infinite dimensions. If (H) has finite dimension (n), then
[ \operatorname{Tr}(I)=n. ]
If (H) is infinite-dimensional, every orthonormal basis contributes infinitely many unit diagonal entries, and therefore
[ \operatorname{Tr}(I)=\infty. ]
Thus the ordinary trace does not define a finite linear functional on the full algebra (\mathcal B(H)) of bounded operators when (H) is infinite-dimensional.
For positive operators (A) and (B), the extended trace satisfies
[ \operatorname{Tr}(A+B) =\operatorname{Tr}(A)+\operatorname{Tr}(B), ]
with addition interpreted in the extended nonnegative real numbers. It is also monotone: if (0\leq A\leq B), then
[ \operatorname{Tr}(A)\leq \operatorname{Tr}(B). ]
These properties make the positive trace analogous to integration of nonnegative measurable functions, where infinite values are admitted before an integrability condition is imposed.
Trace-class operators
For a bounded operator (T) on (H), define its absolute value by
[ |T|=(T^*T)^{1/2}. ]
The operator (T) is trace class when
[ \operatorname{Tr}(|T|)<\infty. ]
The quantity
[ |T|_1=\operatorname{Tr}(|T|) ]
is the trace norm. The set (\mathcal S_1(H)) of trace-class operators is a Banach space and a two-sided ideal in (\mathcal B(H)). In particular, if (T) is trace class and (A) is bounded, then both (AT) and (TA) are trace class, with
[ |AT|_1\leq |A||T|_1, \qquad |TA|_1\leq |A||T|_1. ]
For (T\in\mathcal S_1(H)), the trace is defined by
[ \operatorname{Tr}(T) =\sum_j\langle Te_j,e_j\rangle. ]
This series converges absolutely and has the same value for every orthonormal basis. The resulting map
[ \operatorname{Tr}:\mathcal S_1(H)\longrightarrow\mathbb C ]
is a continuous linear functional satisfying
[ |\operatorname{Tr}(T)|\leq |T|_1. ]
If (A) is bounded and (T) is trace class, cyclicity takes the form
[ \operatorname{Tr}(AT)=\operatorname{Tr}(TA). ]
The restriction that one factor be trace class is essential. Two bounded operators can have products whose diagonal series fail to converge, and formal rearrangement of such series does not define a trace.
Every trace-class operator is compact. If (T) is trace class and ((\lambda_j)) is its sequence of eigenvalues counted with algebraic multiplicity, Lidskii’s theorem gives
[ \operatorname{Tr}(T)=\sum_j\lambda_j. ]
The eigenvalue series is absolutely convergent. This theorem is the infinite-dimensional counterpart of the finite-dimensional statement that the trace equals the sum of the eigenvalues.
Singular values and related operator ideals
The singular values of a compact operator (T) are the eigenvalues of (|T|), written in nonincreasing order and repeated according to multiplicity. A compact operator is trace class precisely when
[ \sum_{j=1}^{\infty}s_j(T)<\infty. ]
This characterization identifies the trace norm with the (\ell^1)-norm of the singular-value sequence:
[ |T|1=\sum{j=1}^{\infty}s_j(T). ]
More generally, the Schatten class (\mathcal S_p(H)) consists of compact operators whose singular values belong to (\ell^p). The case (p=2) gives the Hilbert–Schmidt operators, for which
[ |T|_2^2=\operatorname{Tr}(T^*T). ]
Products of two Hilbert–Schmidt operators are trace class. Consequently, if (A) and (B) are Hilbert–Schmidt, then (\operatorname{Tr}(AB)) is defined and satisfies the corresponding cyclic identity.
Unitary invariance and tensor products
If (U) is unitary and (T) is trace class, then
[ \operatorname{Tr}(UTU^*)=\operatorname{Tr}(T). ]
This is the infinite-dimensional form of invariance under a change of orthonormal coordinates. It also shows that the trace depends on the operator itself rather than on a particular matrix representation.
For trace-class operators (A) on (H) and (B) on (K), their tensor product is trace class on the Hilbert-space tensor product, and
[ \operatorname{Tr}_{H\otimes K}(A\otimes B) =\operatorname{Tr}_H(A)\operatorname{Tr}_K(B). ]
This multiplicativity supports the definition of the partial trace. If (T) is trace class on (H\otimes K), the partial trace over (K) is the unique trace-class operator (\operatorname{Tr}_K(T)) on (H) satisfying
[ \operatorname{Tr}_H!\left(A,\operatorname{Tr}_K(T)\right)
\operatorname{Tr}_{H\otimes K}!\left((A\otimes I_K)T\right) ]
for every bounded operator (A) on (H). Unlike the scalar-valued full trace, the partial trace is an operator-valued map.
Traces on operator algebras
Within a von Neumann algebra (M), a trace is a positive functional or positive extended-valued map satisfying
[ \tau(x^x)=\tau(xx^). ]
When products are within its finite domain, this condition yields
[ \tau(xy)=\tau(yx). ]
A trace is faithful when (\tau(x^*x)=0) implies (x=0), and it is normal when it preserves suprema of increasing bounded families of positive operators. Semifiniteness requires sufficiently many positive elements to have finite trace, allowing finite-trace elements to approximate the positive part of the algebra.
On the full matrix algebra (M_n(\mathbb C)), every tracial linear functional is a scalar multiple of the ordinary matrix trace. On a finite factor, the normalized trace is determined by the condition
[ \tau(I)=1. ]
For the bounded operators on an infinite-dimensional separable Hilbert space, the canonical semifinite trace agrees with the usual operator trace on positive trace-class elements, while assigning infinite value to the identity. Other von Neumann algebras can possess finite traces even when their underlying Hilbert-space representations are infinite-dimensional, a feature central to the classification of factors.
Relation to quantum theory
A density operator is a positive trace-class operator (\rho) satisfying
[ \operatorname{Tr}(\rho)=1. ]
For a bounded observable (A), the associated expectation value is
[ \langle A\rangle_\rho=\operatorname{Tr}(\rho A). ]
Cyclicity ensures that equivalent trace-class products yield the same scalar expectation. For a composite system, the reduced density operator is obtained through the partial trace, which preserves positivity and total trace while discarding the operator degrees of freedom associated with one tensor factor.
The trace also enters the definition of the von Neumann entropy,
[ S(\rho)=-\operatorname{Tr}(\rho\log\rho), ]
whenever the positive operator (-\rho\log\rho) has a well-defined trace. In spectral form, this expression is the sum of (-\lambda_j\log\lambda_j) over the eigenvalues of (\rho).