We consider a class of abstract evolution problems characterized by the sum of two unbounded linear operators $A$ and $B$, where $A$ is assumed to generate a positive semigroup of contractions on an $L^1$-space and $B$ is positive. We study the relations between the semigroup generator $G$ and the operator $A + B$. $A$ characterization theorem for $G =A+B$ is stated. The results are based on the spectral analysis of $B(\lambda-A)^{-1}$. The main point is to study the conditions under which the value 1 belongs to the resolvent set, the continuous spectrum, or the residual spectrum of $B(\lambda - A)^{-1}$.

### A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators

#### Abstract

We consider a class of abstract evolution problems characterized by the sum of two unbounded linear operators $A$ and $B$, where $A$ is assumed to generate a positive semigroup of contractions on an $L^1$-space and $B$ is positive. We study the relations between the semigroup generator $G$ and the operator $A + B$. $A$ characterization theorem for $G =A+B$ is stated. The results are based on the spectral analysis of $B(\lambda-A)^{-1}$. The main point is to study the conditions under which the value 1 belongs to the resolvent set, the continuous spectrum, or the residual spectrum of $B(\lambda - A)^{-1}$.
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semigroup theory; perturbation theory; abstract Cauchy problem
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11584/13079
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