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Perturbations of bi-continuous semigroups. (English) Zbl 1069.47046
Let $$X$$ be a Banach space endowed with another coarser locally convex topology $$\tau$$ such that $$\| x\| =\sup\{| \phi(x):\;\phi\in (X,\tau)',\;\| \phi\| '\leq 1\}$$ for all $$x\in X$$, and $$(X,\tau)'$$ is sequentially complete on $$\tau$$-closed, norm bounded sets. A semigroup $$(T(t))_{t\geq 0}$$ of bounded linear operators on $$(X,\|\;\|)$$ is said to be bi-continuous if (i) $$\sup_{t\in [0,T]}\| T(t)\| <\infty$$ for all $$T>0$$, (ii) $$\tau-\lim_{n\to \infty}T(t)x_n=0$$ uniformly for $$t\in [0,T]$$ for all $$T>0$$ and for every norm-bounded $$\tau$$-null sequence $$(x_n)_n$$, and (iii) the maps $$t\to T(t)x$$ are $$\tau$$-continuous. The notion of bi-continuous semigroups was introduced and studied in [F. Kühnemund, “Bi-continuous semigroups on spaces with two topologies: theory and applications” (Ph. D. Thesis, Tübingen) (2001); see also F. Kühnemund, Semigroup Forum 67, 205–255 (2003)].
In the paper under review, the author provides a bounded perturbation theorem for bi-continuous semigroups. Precisely, it is shown that if $$B\in L(X,\|\;\|)$$ is sequentially $$\tau$$-continuous on norm-bounded sets and $$(A,D(A))$$ is the generator of a bi-continuous semigroup $$(T(t))_{t\geq 0}$$ on $$X$$, then $$(A+B,(D(A))$$ is also the generator of a bi-continuous semigroup $$(S(t))_{t\geq 0}$$ on $$X$$, given by the Dyson-Phillips series. Further, the author gives some examples showing that a bounded perturbation theorem of bi-continuous semigroups does not hold in general.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47A55 Perturbation theory of linear operators 46A03 General theory of locally convex spaces
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