论文标题
在有序的Banach空间上的积极半群的稳定性标准
Stability criteria for positive semigroups on ordered Banach spaces
论文作者
论文摘要
我们考虑了$ C_0 $ -Semigroups的发电机，更普遍地是在有订购的Banach空间上解析阳性运营商$ a $ a $，并寻求确保其光谱约束$ s（a）$的消极情绪的条件。我们的主要结果是$ s（a）<0 $在所谓的\ emph {小增益条件}方面描述了$ ax $ $ x $的行为$ x $。即使在基础空间为$ l^p $ - 空间或连续功能的空间时，这也是新的。 我们还证明，如果基础Banach空间的锥体具有非空的内部或$ a $的基本频谱为负，则表征属性$ s（a）<0 $变得更加容易。为了治疗后一种案例，我们讨论了针对分辨率运算符的克雷因·鲁特曼定理的同行。当$ a $是积极$ C_0 $ - 元素的生成器时，我们的结果可以解释为Semigroup的稳定性结果，因此，它们在离散时间案例中最近证明了相似的结果。 同样，我们证明了一个collatz-Wielandt类型的公式和一个对数公式，用于阳性半群发电机的光谱。
We consider generators of positive $C_0$-semigroups and, more generally, resolvent positive operators $A$ on ordered Banach spaces and seek for conditions ensuring the negativity of their spectral bound $s(A)$. Our main result characterizes $s(A) < 0$ in terms of so-called \emph{small-gain conditions} that describe the behaviour of $Ax$ for positive vectors $x$. This is new even in case that the underlying space is an $L^p$-space or a space of continuous functions. We also demonstrate that it becomes considerably easier to characterize the property $s(A) < 0$ if the cone of the underlying Banach space has non-empty interior or if the essential spectral bound of $A$ is negative. To treat the latter case, we discuss a counterpart of a Krein-Rutman theorem for resolvent positive operators. When $A$ is the generator of a positive $C_0$-semigroup, our results can be interpreted as stability results for the semigroup, and as such, they complement similar results recently proved for the discrete-time case. In the same vein, we prove a Collatz--Wielandt type formula and a logarithmic formula for the spectral bound of generators of positive semigroups.
