A note on approximation of operator semigroups

TL;DR

This paper investigates the asymptotic behavior of operator semigroups generated by bounded linear operators with a large multiple of a projection, providing explicit limit semigroups as the scalar tends to infinity.

Contribution

It introduces a method to analyze the convergence of semigroups involving a large multiple of a projection and explicitly computes the limit semigroup.

Findings

Semigroup $(e^{t(A-kP)})$ converges as $k o abla$

Limit semigroup is explicitly characterized

Convergence occurs on a subspace of the Banach space

Abstract

Let $A$ be a bounded linear operator and $P$ a bounded linear projection on a Banach space $X$. We show that the operator semigroup $(e^{t(A-kP)})_{t \ge 0}$ converges to a semigroup on a subspace of $X$ as $k \to \infty$ and we compute the limit semigroup.