## 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

## Abstract:

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 $t$.## References

- Bernard Aupetit and Driss Drissi,
*Conformal transformations of dissipative operators*, Math. Proc. R. Ir. Acad.**99A**(1999), no. 2, 141–153. MR**1881805** - Nour-Eddine Benamara and Nikolai Nikolski,
*Resolvent tests for similarity to a normal operator*, Proc. London Math. Soc. (3)**78**(1999), no. 3, 585–626. MR**1674839**, DOI 10.1112/S0024611599001756 - N. Borovykh and M. N. Spijker,
*Bounding partial sums of Fourier series in weighted $L^2$-norms, with applications to matrix analysis*, J. Comput. Appl. Math.**147**(2002), no. 2, 349–368. MR**1933601**, DOI 10.1016/S0377-0427(02)00441-7 - Jan A. van Casteren,
*Operators similar to unitary or selfadjoint ones*, Pacific J. Math.**104**(1983), no. 1, 241–255. MR**683741**, DOI 10.2140/pjm.1983.104.241 - Ruth F. Curtain and Hans Zwart,
*An introduction to infinite-dimensional linear systems theory*, Texts in Applied Mathematics, vol. 21, Springer-Verlag, New York, 1995. MR**1351248**, DOI 10.1007/978-1-4612-4224-6 - J. L. M. van Dorsselaer, J. F. B. M. Kraaijevanger, and M. N. Spijker,
*Linear stability analysis in the numerical solution of initial value problems*, Acta numerica, 1993, Acta Numer., Cambridge Univ. Press, Cambridge, 1993, pp. 199–237. MR**1224683**, DOI 10.1017/S0962492900002361 - Tatjana Eisner,
*Polynomially bounded $C_0$-semigroups*, Semigroup Forum**70**(2005), no. 1, 118–126. MR**2107198**, DOI 10.1007/s00233-004-0151-z - Klaus-Jochen Engel and Rainer Nagel,
*One-parameter semigroups for linear evolution equations*, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR**1721989** - Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden,
*Weighted norm inequalities for the conjugate function and Hilbert transform*, Trans. Amer. Math. Soc.**176**(1973), 227–251. MR**312139**, DOI 10.1090/S0002-9947-1973-0312139-8 - J. F. B. M. Kraaijevanger,
*Two counterexamples related to the Kreiss matrix theorem*, BIT**34**(1994), no. 1, 113–119. MR**1429692**, DOI 10.1007/BF01935020 - Heinz-Otto Kreiss,
*Über Matrizen die beschränkte Halbgruppen erzeugen*, Math. Scand.**7**(1959), 71–80 (German). MR**110952**, DOI 10.7146/math.scand.a-10563 - Heinz-Otto Kreiss,
*Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren*, Nordisk Tidskr. Informationsbehandling (BIT)**2**(1962), 153–181 (German, with English summary). MR**165712**, DOI 10.1007/bf01957330 - Randall J. LeVeque and Lloyd N. Trefethen,
*On the resolvent condition in the Kreiss matrix theorem*, BIT**24**(1984), no. 4, 584–591. MR**764830**, DOI 10.1007/BF01934916 - Christian Lubich and Olavi Nevanlinna,
*On resolvent conditions and stability estimates*, BIT**31**(1991), no. 2, 293–313. MR**1112225**, DOI 10.1007/BF01931289 - C. A. McCarthy and J. Schwartz,
*On the norm of a finite Boolean algebra of projections, and applications to theorems of Kreiss and Morton*, Comm. Pure Appl. Math.**18**(1965), 191–201. MR**180867**, DOI 10.1002/cpa.3160180118 - N. Kalton, S. Montgomery-Smith, The paper by Spijker, Tracogna and Welfert, 2004.
- Jan Prüss,
*On the spectrum of $C_{0}$-semigroups*, Trans. Amer. Math. Soc.**284**(1984), no. 2, 847–857. MR**743749**, DOI 10.1090/S0002-9947-1984-0743749-9 - M. N. Spijker, S. Tracogna, and B. D. Welfert,
*About the sharpness of the stability estimates in the Kreiss matrix theorem*, Math. Comp.**72**(2003), no. 242, 697–713. MR**1954963**, DOI 10.1090/S0025-5718-02-01472-2 - John C. Strikwerda and Bruce A. Wade,
*A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions*, Linear operators (Warsaw, 1994) Banach Center Publ., vol. 38, Polish Acad. Sci. Inst. Math., Warsaw, 1997, pp. 339–360. MR**1457017** - H. Zwart, On the estimate $\|(sI-A)^{-1}\| \leq M/\mathrm {Re}(s)$,
*Ulmer Semenaire 2003*, pp. 384–388, 2003.

## Additional Information

**Tatjana Eisner**- Affiliation: Arbeitsbereich Funktionalanalysis, Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
- Email: talo@fa.uni-tuebingen.de
**Hans Zwart**- Affiliation: Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
- Email: h.j.zwart@math.utwente.nl
- Received by editor(s): March 14, 2005
- Received by editor(s) in revised form: September 13, 2005
- Published electronically: July 10, 2006
- © 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: https://doi.org/10.1090/S0025-5718-06-01862-X
- MathSciNet review: 2240644

Dedicated: Dedicated to M.N. Spijker on the occasion of his 65th birthday.