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. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators.

In quantum mechanics, quantum states are described by density matrices, which are certain trace class operators.

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 use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces).

Definition

be a separable Hilbert space an orthonormal basis a positive bounded linear operator. The trace and defined as

is called trace class if and only if

denotes the positive-semidefinite Hermitian square root.

The trace-norm of a trace class operator One can show that the trace-norm is a norm, with the trace-norm, becomes a Banach space.

this definition coincides with that of the trace of a matrix is always self-adjoint) though the converse is not necessarily true.

Equivalent formulations

being in the trace class:

• is finite for every orthonormal basis of .

• is a nuclear operator.

  1. There exist two orthogonal and positive real numbers in \ell^1 and

  2. are the singular values of ), with each value repeated as often as its multiplicity.

  is a compact operator

  1. If is trace class then

  is an integral operator.

• is equal to the composition of two Hilbert-Schmidt operators.

• is a Hilbert-Schmidt operator.

Examples

Spectral theorem

is trace class if and only if has a pure point spectrum such that

Mercer's theorem

Mercer's theorem is a continuous symmetric positive-definite kernel, defined as

then the associated Hilbert–Schmidt integral operator is trace class, i.e.,

Finite-rank operators

Every finite-rank operator is a trace-class operator. Furthermore, the space of all finite-rank operators is a dense subspace (when endowed with the trace norm).

is a continuous linear operator of rank 1 and is thus trace class; moreover, for any bounded linear operator A on H (and into H

Properties

is a non-negative self-adjoint operator is trace-class if and only if are both trace-class. (The positive and negative parts of a self-adjoint operator are obtained by the continuous functional calculus.)

The trace is a linear functional is an inner product on the trace class; the corresponding norm is called the Hilbert–Schmidt norm. The completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.

are also trace-class (i.e. the space of trace-class operators on H is a two-sided ideal in the algebra of bounded linear operators on H), and Furthermore, under the same hypothesis, The last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt.

are two orthonormal bases of H and if T

If A is trace-class, then one can define the Fredholm determinant is invertible.

is trace class then for any orthonormal basis is finite.

for some Hilbert-Schmidt operators holds.

Lidskii's theorem

). Lidskii's theorem (named after Victor Borisovich Lidskii

Note that the series on the right converges absolutely due to Weyl's inequality and the singular values See also Grothendieck trace theorem.

Relationship between common classes of operators

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 the compact operators and finite-rank operators (the sequences that have only finitely many non-zero terms). To some extent, the relationships between these classes of operators are similar to the relationships between their commutative counterparts.

Also, the usual operator norm

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

Trace class as the dual of compact operators

The dual space

is trace-class. An appeal to polar decomposition need not be positive.

is isometrically isomorphic

As the predual of bounded operators

is a two-sided ideal we may define a continuous linear functional of the dual space is an isometric isomorphism This can be used to define the weak-* topology

See also

References

Bibliography

• Dixmier, J. (1969). Les Algebres d'Operateurs dans l'Espace Hilbertien. Gauthier-Villars.