Trace-Class Operators — Mathematics Knowledge System

Overview

A compact operator T on a Hilbert space H is trace-class (or nuclear) if the series of singular values is summable: sum_{n=1}^inf sigma_n(T) < infinity, where sigma_n are the eigenvalues of |T| = (T*T)^{1/2} in decreasing order. Trace-class operators form a two-sided ideal B_1(H) in B(H), and any trace-class operator has a well-defined trace tr(T) = sum_n <Te_n, e_n> independent of the orthonormal basis {e_n}. These operators appear in quantum mechanics as density matrices.

Intuition

The singular values of a compact operator measure its 'size' in each orthogonal direction. For trace-class operators, these sizes are summable, meaning the operator is 'absolutely convergent' in a matrix sense. The trace is the natural extension of the matrix trace: for an infinite matrix, we sum the diagonal entries, and trace-class ensures this sum converges absolutely regardless of the chosen basis.

Formal Definition

Definition

For T in B(H), define |T| = (T*T)^{1/2} (the positive square root of T*T). The singular values are the eigenvalues sigma_n(T) of |T| in decreasing order. T is trace-class if ||T||_1 = sum_{n=1}^inf sigma_n(T) < infinity. The trace-class operators B_1(H) form a Banach space with norm ||·||_1. For T in B_1(H) and any orthonormal basis {e_n}, the trace is tr(T) = sum_n <Te_n,e_n>, which converges absolutely and is independent of the basis.

\|T\|_1 = \sum_{n=1}^\infty \sigma_n(T) = \operatorname{tr}(|T|) < \infty

Trace norm

\operatorname{tr}(T) = \sum_{n=1}^\infty \langle T e_n, e_n \rangle

Trace via orthonormal basis

\|T\|_1 = \sup\left\{ \left|\sum_n \langle T e_n, f_n \rangle\right| : \{e_n\},\{f_n\} \text{ ONS} \right\}

Variational formula for trace norm

Notation

Notation	Meaning
$\mathcal{B}_1(H)$	Banach space of trace-class operators on H
$\\|T\\|_1$	Trace norm (sum of singular values)
$\operatorname{tr}(T)$	Trace of T
$\sigma_n(T)$	n-th singular value of T

Theorems

Theorem 7.1: Theorem 7.1

B_1(H) is a two-sided ideal in B(H): if T in B_1(H) and A in B(H), then AT and TA are in B_1(H) with ||AT||_1 <= ||A|| ||T||_1. Moreover, tr(AB) = tr(BA) for A in B_1(H) and B in B(H).

Theorem 7.2: Theorem 7.2 (Duality)

The dual of B_1(H) (with trace norm) is isometrically isomorphic to B(H) (with operator norm), via the pairing <T, A> = tr(TA) for T in B_1(H), A in B(H). The predual of B(H) is the compact operators K(H).

Theorem 7.3: Theorem 7.3

For a positive trace-class operator T >= 0, tr(T) = sum_n sigma_n(T) = sum_n lambda_n(T) where lambda_n are the eigenvalues of T. For general trace-class T, tr(T) = sum_n lambda_n(T) (counting multiplicity) where the sum converges absolutely.

Worked Examples

1
Since Te_n = a_n e_n >= 0, T is a positive operator. Its singular values are sigma_n(T) = a_n.
2
By definition, T is trace-class iff sum sigma_n = sum a_n < infinity.
3
If sum a_n < infinity, tr(T) = sum_n <Te_n,e_n> = sum_n a_n.
$\operatorname{tr}(T) = \sum_{n=1}^\infty a_n$
4
The basis independence is clear here since the basis {e_n} diagonalises T.

✓ Answer

The diagonal operator with non-negative entries a_n is trace-class iff sum a_n < infinity, and then tr(T) = sum a_n.

Practice Problems

Mediumproof writing

Show that every trace-class operator is compact.

Hardfree response

In quantum mechanics, the expectation value of observable A in state rho is tr(rho A). Explain why rho must be trace-class and positive with tr(rho) = 1.

Quiz

Summary

T is trace-class if sum sigma_n(T) < infinity, where sigma_n are the singular values of T.
The trace norm ||T||_1 = sum sigma_n makes B_1(H) a Banach space and two-sided ideal in B(H).
The trace tr(T) = sum <Te_n,e_n> is independent of the orthonormal basis.
Trace cyclicity: tr(AB) = tr(BA) for A trace-class and B bounded.
Every trace-class operator is compact (its singular values tend to 0 and are summable).

References

BookSimon, B. — Trace Ideals and Their Applications (2nd ed.), AMS, 2005
BookReed, M. & Simon, B. — Methods of Modern Mathematical Physics I, Chapter VI
Websiteen.wikipedia.org