Coherent local hyperbolicity of a linear extension and the essential spectra of a weighted shift operator on a closed interval

In the spaces LP of vector functions on a closed interval, weighted shift operators B generated by diffeomorphisms of the interval are considered. The notion of coherent local hyperbolicity of the associated linear extension is introduced, and it is established that the closedness of the range of the operator I - B is equivalent to coherent local hyperbolicity. On the basis of this result, the description of some essential spectra of the operator B is given.