Defect numbers of singular integral operators with Carleman shift and almost periodic coefficients (Q645369)

This paper is devoted to the study of properties of the kernel and the cokernel of singular integral operators \(T=AP+IQ:L^2({\mathbb R})\to L^2({\mathbb R})\), where \(I\) denotes the identity operator, \(P={1\over 2}(I+S)\), \(Q={1\over 2}(I-S)\), \((S\varphi)(x)={1\over \pi i}\int_{\mathbb R} {\varphi(\tau) \over \tau-x} d\tau\), \(A=\phi_1 I+\phi_2 J\) with almost periodic elements \(\phi_1\) and \(\phi_2\), \(J\) is the reflection operator \((Jf)(x)=f(-x)\). In particular, the dimensions of their kernels and cokernels are obtained. This is done by considering appropriate properties of the related almost periodic elements and, in special, the partial indices of some of their relevant factorizations.