  In this work we generalize some of the previously known adiabatic theorems to
situations with non-unitary evolutions in Banach spaces. We prove adiabatic
theorems with uniform gap condition (generalizing a theorem of Abou Salem),
adiabatic theorems with non-uniform gap condition (generalizing a theorem of
Kato) and qualitative as well as quantitative adiabatic theorems without gap
condition (generalizing theorems of Avron and Elgart, and Teufel).
Additionally, we give a generalized version of an adiabatic theorem of higher
order due to Nenciu. In all these adiabatic theorems the considered spectral
values need not lie on the imaginary axis and in the adiabatic theorems with
spectral gap condition and the adiabatic theorem of higher order compact
subsets of the spectrum are sufficient (in particular, these subsets need not
consist of eigenvalues). We explore the strength of the presented adiabatic
theorems in numerous examples. In particular, we show that the theorems of the
present work are more general than the previously known theorems.
  This work was finished in October 2010 and handed in as a diploma thesis at
the mathematics department of the University of Stuttgart. The more recent
results of Avron, Fraas, Graf und Grech which appeared in the meantime (in June
2011) are therefore not taken into consideration here. We point out, however,
that the theorems of the present work are more general than the corresponding
theorems of Avron, Fraas, Graf und Grech - with the exception of the adiabatic
theorem of higher order which is in no logical relation to the adiabatic
theorem of higher order of Avron, Fraas, Graf and Grech.
  We are about to gather the most important theorems of the present work in an
article and will upload it to arXiv soon.
