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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Continuous-time Kreiss resolvent condition on infinite-dimensional spaces
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by Tatjana Eisner and Hans Zwart PDF
Math. Comp. 75 (2006), 1971-1985 Request permission


Given the infinitesimal generator $A$ of a $C_0$-semigroup on the Banach space $X$ which satisfies the Kreiss resolvent condition, i.e., there exists an $M>0$ such that $\| (sI-A)^{-1}\| \leq \frac {M}{\mathrm {Re}(s)}$ for all complex $s$ with positive real part, we show that for general Banach spaces this condition does not give any information on the growth of the associated $C_0$-semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow at most like $t$. Furthermore, we show that for every $\gamma \in (0,1)$ there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows at least like $t^\gamma$. As a consequence, we find that for ${\mathbb R}^N$ with the standard Euclidian norm the estimate $\|\exp (At)\| \leq M_1 \min (N,t)$ cannot be replaced by a lower power of $N$ or $t$.
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Additional Information
  • Tatjana Eisner
  • Affiliation: Arbeitsbereich Funktionalanalysis, Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
  • Email:
  • Hans Zwart
  • Affiliation: Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
  • Email:
  • Received by editor(s): March 14, 2005
  • Received by editor(s) in revised form: September 13, 2005
  • Published electronically: July 10, 2006

  • Dedicated: Dedicated to M.N. Spijker on the occasion of his 65th birthday.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 75 (2006), 1971-1985
  • MSC (2000): Primary 47D06, 15A60; Secondary 65J10, 34K20, 47N40
  • DOI:
  • MathSciNet review: 2240644