I have been the paper titled Dual Piecewise Analytic Bundle Shift Models of Linear Operators by Dmitry Yakubovich.

In the second paragraph of the introduction it says "Let $T$ be a bounded Linear operator on a reflexive Banach space $X$. It will be assumed that *$T$ behaves like the shift operator, that is, that the eigenvalues of $T^*$ fill in some connected components of the complement of the essential spectrum of $T$, whereas the point spectrum of $T$ is empty.*

Can anyone explain why the statement in italics would be true? Why an operator which behaves like a shift operator have such properties? Any reference to theory related to this would be helpful.