The semi-Fredholm spectra of operators ofthe form $ (Tf)(z)=w(z)f\bigl{(}B(z)\bigr{)} $ acting on the disk algebraare described in the case when $ B $ is either an elliptic or a double parabolic finite Blaschke productof degree $ d\geq 2 $ , and $ w $ has no zeros on the unit circle.When $ B $ has zeros on the unit circle, we formulatea conjecture concerning the essential spectra ofthe above operators and prove some results that partially justify this conjecture.Our results suggest the possibility of interesting connections betweenthe spectral properties of weighted composition operators and complex dynamics.
