In mathematics, the commutator subspace of a two-sided ideal of bounded linear operators on a separable Hilbert space is the linear subspace spanned by commutators of operators in the ideal with bounded operators.Modern characterisation of the commutator subspace is through the Calkin correspondence and it involves the invariance of the Calkin sequence space of an operator ideal to taking Cesàro means. This explicit spectral characterisation reduces problems and questions about commutators and traces on two-sided ideals to (more resolvable) problems and conditions on sequence spaces.
Commutators of linear operators on Hilbert spaces came to prominence in the 1930s as they featured in the matrix mechanics, or Heisenberg, formulation of quantum mechanics. Commutator subspaces, though, received sparse attention until the 1970s. American mathematician Paul Halmos in 1954 showed that every bounded operator on a separable infinite dimensional Hilbert space is the sum of two commutators of bounded operators.[1] In 1971 Carl Pearcy and David Topping revisited the topic and studied commutator subspaces for Schatten ideals.[2] As a student American mathematician Gary Weiss began to investigate spectral conditions for commutators of Hilbert–Schmidt operators.[3] [4] British mathematician Nigel Kalton, noticing the spectral condition of Weiss, characterised all trace class commutators.[5] Kalton's result forms the basis for the modern characterisation of the commutator subspace.In 2004 Ken Dykema, Tadeusz Figiel, Gary Weiss and Mariusz Wodzicki published the spectral characterisation of normal operators in the commutator subspace for every two-sided ideal of compact operators.[6]
The commutator subspace of a two-sided ideal J of the bounded linear operators B(H) on a separable Hilbert space H is the linear span of operators in J of the form [''A'',''B''] = AB - BA for all operators A from J and B from B(H).
The commutator subspace of J is a linear subspace of J denoted by Com(J) or [''B''(''H''),''J''].
The Calkin correspondence states that a compact operator A belongs to a two-sided ideal J if and only if the singular values μ(A) of A belongs to the Calkin sequence space j associated to J. Normal operators that belong to the commutator subspace Com(J) can characterised as those A such that μ(A) belongs to j and the Cesàro mean of the sequence μ(A) belongs to j.[6] The following theorem is a slight extension to differences of normal operators[7] (setting B 0 in the following gives the statement of the previous sentence).
Theorem. Suppose A,B are compact normal operators that belong to a two-sided ideal J. Then A - B belongs to the commutator subspace Com(J) if and only if
\left\{
| 1 | |
| 1+n |
| n | |
| \sum | |
| k=0 |
\left(\mu(k,A)-\mu(k,B)\right)
| infty | |
| \right\} | |
| n=0 |
\inj
where j is the Calkin sequence space corresponding to J and μ(A), μ(B) are the singular values of A and B, respectively.
Provided that the eigenvalue sequences of all operators in J belong to the Calkin sequence space j there is a spectral characterisation for arbitrary (non-normal) operators. It is not valid for every two-sided ideal but necessary and sufficient conditions are known. Nigel Kalton and American mathematician Ken Dykema introduced the condition first for countably generated ideals.[8] [9] Uzbek and Australian mathematicians Fedor Sukochev and Dmitriy Zanin completed the eigenvalue characterisation.
Theorem. Suppose J is a two-sided ideal such that a bounded operator A belongs to J whenever there is a bounded operator B in J such that
If the bounded operator A and B belong to J then A - B belongs to the commutator subspace Com(J) if and only if
\left\{
| 1 | |
| 1+n |
| n | |
| \sum | |
| k=0 |
\left(λ(k,A)-λ(k,B)\right)
| infty | |
| \right\} | |
| n=0 |
\inj
where j is the Calkin sequence space corresponding to J and λ(A), λ(B) are the sequence of eigenvalues of the operators A and B, respectively, rearranged so that the absolute value of the eigenvalues is decreasing.
Most two-sided ideals satisfy the condition in the Theorem, included all Banach ideals and quasi-Banach ideals.
See main article: Singular trace.
A trace φ on a two-sided ideal J of B(H) is a linear functional φ:J →
C
Suppose H is a separable infinite dimensional Hilbert space.
\left\{
| a1+a2+ … +an | |
| n |
| infty | |
| \right\} | |
| n=1 |
\inc00
occurs if and only if
a1+a2+ … +aN=0
for the sequence (a1, a2, ..., aN, 0, 0, ...) in c00. The kernel of the operator trace Tr on F(H) and the commutator subspace of the finite rank operators are equal, ker Tr Com(F(H)) ⊊ F(H).
\left\{
| a1+a2+ … +an | |
| n |
| infty | |
| \right\} | |
| n=1 |
\in\ell1
is stronger than the condition that a1 + a2 ... = 0. An example is the sequence with
an=
| 1 | |
| nlog2(n) |
, n\geq2.
and
a1=-
| infty | |
| \sum | |
| n=2 |
an.
\left\{
| a1+a2+ … +an | |
| n |
| infty | |
| \right\} | |
| n=1 |
\in\ell1,infty
or equivalently
\left\{a1+a2+ … +an
| infty | |
| \right\} | |
| n=1 |
=O(1)