Growth bound for the shift semigroup on weighted Sobolev spaces

摘要

We present sufficient and necessary conditions for the strong continuity of the shift semigroup on weighted Sobolev spaces explicitly in terms of the weight function. Further, we discuss properties of the spectrum and the point spectrum of the generator of the semigroup. We show how the spectrum and point spectrum of the generator and the growth bound of the semigroup depend on the weight function.