In Krengel's it's argued that the fact that $\exists 0\ne u \in L_\infty$ orthogonal to $(zI-T)L_1$ , where $z$ is a complex number on the unit circle, $|z|=1$, then $T^* u = zu$.
I don't understnad how did they get: $T^* u = zu$?
I mean $(u,(zI-T)v)=0 \forall v \in L^1$ which would entail that $T^* u = \bar{z}u= 1/z u$, since $z\bar{z}=1$.
It's presented on page 317 in the proof of proposition 25. https://books.google.co.il/books?id=OAUyQph3oVYC&printsec=frontcover#v=onepage&q&f=false
By the definition of the adjoint you have for all $v\in L^1$ $$T^*(u)(v)= u(T(v))=u(zv)=zu(v),$$ hence $T^*(u)=zu$.