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

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We establish adiabatic theorems with and without spectral gap condition for general operators A(t): D(A(t)) ⊂ X → 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 σ(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 σ(t) = { λ(t) } consist of eigenvalues λ(t) ∈ ∂ σ(A(t)) and that there exist projections P(t) reducing A(t) such that A(t)|ₚ₍ₜ₎D₍ₐ₍ₜ₎₎-λ(t) is nilpotent and A(t)|₍₁₋ₚ₍ₜ₎₎D₍ₐ₍ₜ₎₎-λ(t) is injective with dense range in (1-P(t))X for almost every t. In all these theorems, the regularity conditions imposed on t A(t), σ(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.