Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces,II

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator (mathscr {L}= -mathrm{d}/mathrm{d}t+A) in homogeneous function spaces. We focus on the dependency between various invertibility states of such an operator. In particular, we prove that often, a generally weaker state of invertibility implies a stronger state for (mathscr {L}) under mild additional conditions. For example, we show that if the operator (mathscr {L}) is surjective and the imaginary axis is not contained in the interior of the spectrum of A, then (mathscr {L}) is invertible.