Trace-class operators and commutators (Q1825424)

If I and J are two-sided ideals in the algebra of bounded operators B on \(\ell_ 2\), denote by [I,J] the linear span of the commutators AB-BA with \(A\in I\), \(B\in J\). N. J. Kalton gives a beautiful characterization of the trace-class commutators [\({\mathcal S}_ 1,B]\), solving a problem open for some time. For the Schatten p-classes \({\mathcal S}_ p\) and \(p\neq 1\), the corresponding characterization of [\({\mathcal S}_ p,B]\) is known and simpler. For \(p=1\), one gets the linear space \({\mathcal C}{\mathcal S}_ 1\) of those \(T\in {\mathcal S}_ 1\) whose eigenvalues \(\lambda_ n(T)\) satisfy \[ \sum_{n\in {\mathbb{N}}}n^{-1}| \lambda_ 1(T)+...+\lambda_ n(T)| <\infty. \] Hence trace T\(=0\) for \(T\in {\mathcal C}{\mathcal S}_ 1\) (but not conversely). The commutator space [\({\mathcal S}_ o,{\mathcal S}_ q]\) with \(1/p+1/q=1\) also coincides with \({\mathcal C}{\mathcal S}_ 1\). At most 6 commutators are needed to reach any \(T\in {\mathcal C}{\mathcal S}_ 1\); \({\mathcal C}{\mathcal S}_ 1\) is also equal to the linear span of the quasi- nilpotent trace-class operators. The techniques used are extensios of ideas concerning nonlinear commutators arising in interpolation theory. They are also connected with the theory of twisted sums of Banach spaces due to the author.