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

Volume 75 / 2007

Hans Zwart Banach Center Publications 75 (2007), 303-313 MSC: Primary 47D60; Secondary 47D06, 93C25, 93D20, 46N40. DOI: 10.4064/bc75-0-18

Abstract

In this paper we study the question whether $A^{-1}$ is the infinitesimal generator of a bounded $C_0$-semigroup if $A$ generates a bounded $C_0$-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.

Authors

  • Hans ZwartDepartment of Applied Mathematics
    University of Twente
    P.O. Box 217
    7500 AE Enschede
    The Netherlands
    e-mail

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