## Arkiv för Matematik

• Ark. Mat.
• Volume 43, Number 2 (2005), 365-382.

### Distance near the origin between elements of a strongly continuous semigroup

Jean Esterle

#### Abstract

Set $\theta (s/t): = (s/t - 1)(t/s)^{\frac{{s/t}}{{s/t - 1}}} = (s - t)\frac{{t^{t/(s - t)} }}{{s^{s/(s - t)} }}$ if 0< t< s. The key result of the paper shows that if (T (t))t>0 is a nontrivial strongly continuous quasinilpotent semigroup of bounded operators on a Banach space then there exists δ>0 such that ║T(t)-T(s)║>θ(s/t) for 0< t< s≤δ. Also if (T(t))t>0 is a strongly continuous semigroup of bounded operators on a Banach space, and if there exists η>0 and a continuous function ts(t) on [0, ν], satisfying s(0)=0, and such that 0< t< s(t) and ║T(t)-T(s(t))║<θ(s/t) for t∈(o, η], then the infinitesimal generator of the semigroup is bounded. Various examples show that these results are sharp.

#### Note

This work is part of the research program of the network ‘Analysis and operators’, contract HPRN-CT 2000 00116, funded by the European Commission.

#### Article information

Source
Ark. Mat., Volume 43, Number 2 (2005), 365-382.

Dates
Received: 8 January 2004
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898905

Digital Object Identifier
doi:10.1007/BF02384785

Mathematical Reviews number (MathSciNet)
MR2173957

Zentralblatt MATH identifier
1128.47039

Rights
2005 © Institut Mittag-Leffler

#### Citation

Esterle, Jean. Distance near the origin between elements of a strongly continuous semigroup. Ark. Mat. 43 (2005), no. 2, 365--382. doi:10.1007/BF02384785. https://projecteuclid.org/euclid.afm/1485898905

#### References

• Berkani, M., Esterle, J. and Mokhtari, A., Distance entre puissances d'une unité approchée bornée, J. London Math. Soc. 67 (2003), 461–480.
• Blake, M. D., A spectral bound for asymptotically norm-continuous semigroups, J. Operator Theory 45 (2001), 111–130.
• El-Mennaoui, O. and Engel, K.-J., On the characterization of eventually norm continuous semigroups in Hilbert space, Arch. Math. (Basel) 63 (1994), 437–440.
• Engel, K.-J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math. 194, Springer, New York, 2000.
• Esterle, J., Quasimultipliers, representations of H, and the closed ideal problem for commutative radical Banach algebras, in Radical Banach Algebras and Automatic Continuity (Long Beach, CA, 1981), Lecture Notes in Math. 975, pp. 66–162, Springer, Berlin, 1983.
• Esterle, J., Zero-one and zero-two laws for the behaviour of semigroups near the origin, in Banach Algebras and their Applications, Proc. of the Edmonton Conference on Banach Algebras, Contemporary Math. 363, pp. 69–80, Amer. Math. Soc., Providence, RI. 2004.
• Esterle, J. and Mokhtari, A., Distance entre éléments d'un semi-groupe dans une algèbre de Banach, J. Funct. Anal. 195 (2002), 167–189.
• Feller, W., On the generation of unbounded semi-groups of bounded linear operators, Ann. of Math. 58 (1953), 166–174.
• Gowers, W. T. and Maurey, B., The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851–874.
• Hille, E. and Phillips, R. S., Functional Analysis and Semi-Groups, Amer. Math. Soc. Coll. Publ. 31, Amer. Math. Soc., Providence, RI, 1957.
• Kalton, N., Montgomery-Smith, S., Oleskiewicz, K. and Tomilov, Y., Powerbounded operators and related norm estimates, J. London Math. Soc. 70 (2004), 463–478.
• Kato, T., A characterization of holomorphic semigroups. Proc. Amer. Math. Soc. 25 (1970), 495–498.
• Mokhtari, A., Distance entre éléments d'un semi-groupe continu dans une algèbre de Banach, J. Operator Theory 20 (1988), 375–380.
• Pazy, A., Approximations of the identity operator by semigroups of linear operators, Proc. Amer. Math. Soc. 30 (1971), 147–150.
• Räbiger, F. and Ricker, W. J., C0-groups and C0-semigroups of linear operators on hereditarily indecomposable Banach spaces. Arch. Math. (Basel) 66 (1996), 60–70.
• Räbiger, F. and Ricker, W. J., C0-semigroups and cosine families of linear operators in hereditarily indecomposable Banach spaces, Acta Sci. Math. (Szeged) 64 (1998), 697–706.
• Rickart, C. E., General Theory of Banach Algebras. Van Nostrand, Princeton. NJ, 1960.
• You, P. H., Characteristic conditions for a C0-semigroup with continuity in the uniform operator topology for t>0 in Hilbert space. Proc. Amer. Math. Soc. 116 (1992), 991–997.