Perturbations of bi-continuous semigroups.

*(English)*Zbl 1069.47046Let \(X\) 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.

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.

Reviewer: A. Albanese (Lecce)

##### MSC:

47D06 | One-parameter semigroups and linear evolution equations |

47A55 | Perturbation theory of linear operators |

46A03 | General theory of locally convex spaces |