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- W2079301145 abstract "Let ${T(t)}_{tge 0}$ be a $C_0$-semigroup with generator $A$ on a Banach space $X$. Let $x_0in X$ be a fixed element. We prove the following individual stability results. (i) Suppose $X$ is an ordered Banach space with weakly normal closed cone $C$ and assume there exists $t_0ge 0$ such that $T(t)x_0in C$ for all $tge t_0$. If the local resolvent $lambda mapsto ( lambda -A)^{-1} x_0$ admits a bounded analytic extension to the right half-plane ${operatorname {Re}lambda >0}$, then for all $mu in varrho (A)$ and $x^ast in X^ast$ we have [ lim _{tto infty } bigl langle T(t)(mu -A)^{-1} x_0, x^ast bigr rangle = 0.] (ii) Suppose $E$ is a rearrangement invariant Banach function space over $[0,infty )$ with order continuous norm. If $x_0^ast in X^ast$ is an element such that $tmapsto langle T(t)x_0, x_0^ast rangle$ defines an element of $E$, then for all $mu in varrho (A)$ and $beta ge 1$ we have [ lim _{tto infty } bigl langle T(t)(mu -A)^{-beta } x_0, x_0^ast bigr rangle = 0.]" @default.
- W2079301145 created "2016-06-24" @default.
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- W2079301145 date "2002-02-04" @default.
- W2079301145 modified "2023-09-24" @default.
- W2079301145 title "On individual stability of $C_0-$semigroups" @default.
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- W2079301145 doi "https://doi.org/10.1090/s0002-9939-02-06341-4" @default.
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