On the compactness of the hypercomplex commutator in Hölder continuous functions spaces (Q1958679)

The authors consider the communicator of a multiplication operator \(M_a\) and the singular integral operator of Cauchy's type \(S_\gamma\) in a space of Hölder-continuous functions in a simply connected bounded domain in the complex plane with values in a Douglis algebra. Such algebra was introduced by \textit{A. Douglis} [Commun. Pure Appl. Math. 6, 291--298 (1953; Zbl 0050.31902)]. The boundary \(\gamma\) should be a rectifiable Jordan curve positively oriented. The main result is the following: Let \(a\in{\mathcal H}_\phi(\gamma)\) (a subclass of Hölder-continuous functions). Then the commutator \(S_\gamma M_a- M- aS_\gamma\) is a compact operator in \({\mathcal B}_0({\mathcal H}_\phi(\gamma))\). Here \({\mathcal B}_0(X)\) describes the algebra of all continuous linear operators on \(X\).