Recently, Michael Gil’ has provided a condition for integrability of norms of semigroups generated by operators of the form $$A+B$$ A + B where B is bounded, in terms of the norm of the commutator of A and B . We show that his argument, when slightly refined, leads to a bit more general result than the one presented in his paper. We also argue, by examining an example, that the size of the commutator may not be a good indicator of whether the perturbed semigroup has integrable norm or not.

中文翻译：

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关于摄动半群范数的可积性的注记

最近，根据 A 和 B 的交换子的范数，Michael Gil' 提供了由 $$A+B$$ A + B 形式的运算符生成的半群范数的可积性条件，其中 B 是有界的。我们表明，他的论点稍加改进后，会得出比他论文中提出的更一般的结果。我们还认为，通过检查一个例子，交换子的大小可能不是扰动半群是否具有可积范数的一个很好的指标。