Adiabatic theorems with and without spectral gap condition for non-semisimple spectral values

We establish adiabatic theorems with and without spectral gap condition for general operators $A(t):D(A(t))\subset X\to X$ with possibly time-dependent domains in a Banach space $X$. We first prove adiabatic theorems with uniform and non-uniform spectral gap condition (including a slightly extended adiabatic theorem of higher order). In these adiabatic theorems the considered spectral subsets $\sigma (t)$ have only to be compact – in particular, they need not consist of eigenvalues. We then prove an adiabatic theorem without spectral gap condition for not necessarily (weakly) semisimple eigenvalues: in essence, it is only required there that the considered spectral subsets $\sigma(t) = { \lambda(t) }$consistofeigenvalues$\lambda(t) \in \partial \sigma(A(t))$andthatthereexistprojections$P(t)$reducing$A(t)$suchthat$A(t)|{P(t)D(A(t))}-\lambda(t)$isnilpotentand$A(t)|{(1-P(t))D(A(t))}-\lambda(t)$isinjectivewithdenserangein$(1-P(t))X$foralmostevery$t$. In all these theorems, the regularity conditions imposed on $t\mapsto A(t)$, $\sigma (t)$, $P(t)$ are fairly mild. We explore the strength of the presented adiabatic theorems in numerous examples. And finally, we apply the adiabatic theorems for time-dependent domains to obtain – in a very simple way – adiabatic theorems for operators $A(t)$ defined by symmetric sesquilinear forms.