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

Banach Center Publications (2007)

- Volume: 75, Issue: 1, page 303-313
- ISSN: 0137-6934

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topHans 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 -

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