Countable spectrum, transfinite induction and stability

Abstract

We reconsider the contour argument and proof by transfinite induction of the ABLV-Theorem given in [AB88]. But here we use the method to prove a Tauberian Theorem for Laplace transforms which has the ABVL Theorem about stability of a semigroup as corollary and also gives quantitative estimates. It is interesting that considering countable spectrum leads to the same problems Cantor encountered when he tried to prove a uniqueness result for trigonometric series. It led him to invent ordinal numbers and transfinite induction. We explain these connections in the article.