Commutator of possibly unbounded operators - Imported Knowlpedia Content

Definition

Let $A$ and $B$ be linear operators on a Hilbert space $H$ , with possibly proper domains. Their operator commutator is the operator

[A,B]\xi=AB\xi-BA\xi

with natural domain

\operatorname{Dom}([A,B])=\operatorname{Dom}(AB)\cap\operatorname{Dom}(BA),

where $\operatorname{Dom}(AB)=\{\xi\in\operatorname{Dom}(B):B\xi\in \operatorname{Dom}(A)\}$ , and similarly for $BA$ . One may instead specify a common subdomain $E$ on which both products are defined; then the notation $[A,B]|_E$ records that choice. Thus a commutator of densely defined operators is not determined by the formal expression alone: its domain is part of the operator.

Domain discipline

Even when $B$ is a bounded operator, the product $AB$ is defined only on vectors $\xi$ for which $B\xi$ lies in $\operatorname{Dom}(A)$ . Boundedness of $B$ does not by itself imply that $B\operatorname{Dom}(A)\subseteq\operatorname{Dom}(A)$ . Consequently, algebraic identities involving commutators must be checked on a domain invariant under every operator product that occurs. This is the operator-domain convention used in the standard treatment of unbounded operators Reed and Simon, Chapter VIII.

Spectral-triple convention

For a spectral triple $(\mathcal A,H,D)$ , the algebra is represented by bounded operators $\pi(a)$ . The usual convention requires $\pi(a)\operatorname{Dom}(D)\subseteq\operatorname{Dom}(D)$ , so that $[D,\pi(a)]$ is initially defined on all of $\operatorname{Dom}(D)$ . The spectral-triple axiom then asks for this operator to have a bounded extension. A source that writes simply $[D,a]$ is normally suppressing both the representation $\pi$ and this initial domain. Form or quadratic-form commutators are weaker conventions and must be identified explicitly.

Example and warning

On $L^2(S^1)$ , let $D=-i\,d/dx$ with Sobolev domain $H^1(S^1)$ , and let $M_f$ multiply by a smooth function $f$ . Multiplication preserves $H^1(S^1)$ , and on that domain

[D,M_f]=-iM_{f'}.

This calculation is legitimate because the domain invariance is known first. Without it, the same formal subtraction could be defined only on a smaller intersection, possibly one that is not dense.

References

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. Publisher record. Relevant: Chapter VIII on unbounded operators and their domains.
José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. DOI record. Relevant: §10.1 on spectral triples and bounded commutators.