Canonical operator trace - Imported Knowlpedia Content

Definition

Let $H$ be a Hilbert space and let $T$ be a positive bounded operator on $H$ . Its canonical operator trace is

\operatorname{Tr}(T)=\sum_{e\in\mathcal E}\langle Te,e\rangle\in[0,\infty],

where $\mathcal E$ is any orthonormal basis and an arbitrary nonnegative sum means the supremum of its finite partial sums. The value is independent of $\mathcal E$ . If it is finite, $T$ is trace class. For a general trace-class operator $S$ , the series $\operatorname{Tr}(S)=\sum_{e\in\mathcal E}\langle Se,e\rangle$ converges absolutely and is likewise basis independent. Thus the positive trace may take $+\infty$ , while the linear trace is defined on the trace-class ideal.

Relation to singular values

A compact operator $S$ is trace class exactly when

\sum_{n=1}^{\infty}s_n(S)<\infty,

where $s_n(S)$ are its singular values. In that case $\operatorname{Tr}(|S|)=\sum_ns_n(S)$ and $|\operatorname{Tr}(S)|\leq\operatorname{Tr}(|S|)$ . For a positive compact operator, the trace is the sum of its eigenvalues counted with multiplicity Simon, §1.

Tracial and continuity properties

If $S$ is trace class and $A$ is bounded, then both $AS$ and $SA$ are trace class and

\operatorname{Tr}(AS)=\operatorname{Tr}(SA).

The trace is positive, faithful on positive operators, and normal: increasing nets of positive operators have traces increasing to the trace of their supremum. Cyclicity must not be applied indiscriminately to products of unbounded operators; the trace-class hypotheses are what make the displayed identity valid.

Scope and normalizations

On $B(H)$ , this trace assigns value one to every rank-one projection and is the canonical faithful normal semifinite trace. Other operator algebras may carry no trace, one trace, or many traces, and a normalized tracial state is a different notion. In finite dimension, the definition reduces to the ordinary matrix trace. In infinite dimension, $\operatorname{Tr}(I)=+\infty$ , so the canonical trace is not a state.

References

Barry Simon, Trace Ideals and Their Applications, 2nd ed., American Mathematical Society, 2005. DOI record. Relevant: §§1–3 on trace-class operators, singular values, and cyclicity.
Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. Publisher record. Relevant: Chapter VI on trace ideals and the canonical trace.