Is $A^{-1}$ an infinitesimal generator?

Hans Zwart. "Is $A^{-1}$ an infinitesimal generator?." Banach Center Publications 75.1 (2007): 303-313. <http://eudml.org/doc/282174>.

@article{HansZwart2007,
abstract = {In this paper we study the question whether $A^\{-1\}$ is the infinitesimal generator of a bounded C₀-semigroup if A generates a bounded C₀-semigroup. If the semigroup generated by A is analytic and sectorially bounded, then the same holds for the semigroup generated by $A^\{-1\}$. However, we construct a contraction semigroup with growth bound minus infinity for which $A^\{-1\}$ does not generate a bounded semigroup. Using this example we construct an infinitesimal generator of a bounded semigroup for which its inverse does not generate a semigroup. Hence we show that the question posed by deLaubenfels in [13] must be answered negatively. All these examples are on Banach spaces. On a Hilbert space the question whether the inverse of a generator of a bounded semigroup also generates a bounded semigroup still remains open.},
author = {Hans Zwart},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {303-313},
title = {Is $A^\{-1\}$ an infinitesimal generator?},
url = {http://eudml.org/doc/282174},
volume = {75},
year = {2007},
}

TY - JOUR
AU - Hans Zwart
TI - Is $A^{-1}$ an infinitesimal generator?
JO - Banach Center Publications
PY - 2007
VL - 75
IS - 1
SP - 303
EP - 313
AB - In this paper we study the question whether $A^{-1}$ is the infinitesimal generator of a bounded C₀-semigroup if A generates a bounded C₀-semigroup. If the semigroup generated by A is analytic and sectorially bounded, then the same holds for the semigroup generated by $A^{-1}$. However, we construct a contraction semigroup with growth bound minus infinity for which $A^{-1}$ does not generate a bounded semigroup. Using this example we construct an infinitesimal generator of a bounded semigroup for which its inverse does not generate a semigroup. Hence we show that the question posed by deLaubenfels in [13] must be answered negatively. All these examples are on Banach spaces. On a Hilbert space the question whether the inverse of a generator of a bounded semigroup also generates a bounded semigroup still remains open.
LA - eng
UR - http://eudml.org/doc/282174
ER -