Commutators

A commutator is an operator of the form $PQ - QP$, where $P$ and $Q$ are operators on a Hilbert space.

The sum of a compact operator and a non-zero scalar is not a commutator. (Proof: in the quotient of the algebra of all operators by the ideal of compact operators, the unit element is not a commutator, by the Wintner and Wielandt proofs of Problem 230; translated back to operators, this means the identity cannot equal the sum of a commutator and a compact operator, and a fortiori neither can a non-zero scalar plus a compact operator, since dividing by that scalar reduces to the previous case.) The Brown--Pearcy characterization of commutators (see Problem 230) implies that, conversely, on a separable Hilbert space every operator that is not the sum of a non-zero scalar and a compact operator is a commutator; in particular it implies that the identity is a limit of commutators, i.e., that there exist sequences $\left\{ P_n\right\}$ and $\left\{ Q_n\right\}$ of operators such that $\left\| 1-(P_nQ_n-Q_nP_n)\right\| \to 0$ as $n \to \infty$ (see also Problem 235). The result below is a more modest, more easily accessible version of that fact.

If the underlying Hilbert space is finite-dimensional, then it is an easy exercise in linear algebra to prove that $\left\| 1-C\right\| \geq 1$ for all commutators $C$.

A commutator is an operator of the form $PQ - QP$, where $P$ and $Q$ are operators on a Hilbert space. A subspace $\mathbf{M}$ of a Hilbert space $\mathbf{H}$ is called large if $\dim \mathbf{M} = \dim \mathbf{H}$; an operator is said to have a large kernel if its kernel, as a subspace, is large.

The self-commutator of an operator $A$ is the operator $A^{*}A - AA^{*}$.

An operator is called abnormal if no non-zero subspace of the kernel of $A^{*}A - AA^{*}$ reduces $A$ (that is, if it has no normal direct summand).

The word "commutator" occurs in two distinct mathematical contexts. In ring theory it means $PQ - QP$ (an additive commutator); in group theory it means $PQP^{-1}Q^{-1}$ (a multiplicative commutator).

A multiplicative commutator is an operator of the form $PQP^{-1}Q^{-1}$, where $P$ and $Q$ are operators on a Hilbert space. Problem 238 shows that if $\mathbf{H}$ is an infinite-dimensional Hilbert space, then a necessary and sufficient condition that a scalar $\alpha$ acting on $\mathbf{H}$ be a multiplicative commutator is that $\left|\alpha \right| = 1$; the present problem strengthens that result.

The commutator subgroup of a group is the smallest subgroup that contains all elements of the form $PQP^{-1}Q^{-1}$; in other words, it is the subgroup generated by all (multiplicative) commutators. The set of all invertible operators on a Hilbert space is a multiplicative group; in analogy with standard finite-dimensional terminology, it may be called the full linear group of the space.