## The operator $a(x)\frac {d}{dx}$ on Banach space

HTML articles powered by AMS MathViewer

- by Fuyuan Yao
- Proc. Amer. Math. Soc.
**125**(1997), 1027-1032 - DOI: https://doi.org/10.1090/S0002-9939-97-03564-8
- PDF | Request permission

## Abstract:

The operator $a(x)\frac {d}{dx}$ on $C(I)$, where $I$ is an interval contained in the real line, is considered in many places. In this paper, we attempt to reconsider it in the subspace of $C_0(-\infty ,\infty )$ containing all even functions, and show that it generates a strongly continuous semigroup. It is interesting that our main conditions seem contradictory to previous ones. It is due to the symmetry of the functions and the different domain of the operator than usual.## References

- C. J. K. Batty,
*Derivations on compact spaces*, Proc. London Math. Soc. (3)**42**(1981), no. 2, 299–330. MR**607305**, DOI 10.1112/plms/s3-42.2.299 - C. J. K. Batty,
*Derivations on the line and flows along orbits*, Pacific J. Math.**126**(1987), no. 2, 209–225. MR**869776**, DOI 10.2140/pjm.1987.126.209 - B. C. Burch and Jerome A. Goldstein,
*Some boundary value problems for the Hamilton-Jacobi equation*, Hiroshima Math. J.**8**(1978), no. 2, 223–233. MR**481442**, DOI 10.32917/hmj/1206135453 - R. Courant and D. Hilbert,
*Methods of mathematical physics. Vol. II: Partial differential equations*, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. (Vol. II by R. Courant.). MR**0140802** - Ralph deLaubenfels,
*Well-behaved derivations on $C[0,\,1]$*, Pacific J. Math.**115**(1984), no. 1, 73–80. MR**762202**, DOI 10.2140/pjm.1984.115.73 - Jerome A. Goldstein,
*Abstract evolution equations*, Trans. Amer. Math. Soc.**141**(1969), 159–185. MR**247524**, DOI 10.1090/S0002-9947-1969-0247524-5 - Jerome A. Goldstein,
*Semigroups of linear operators and applications*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR**790497** - A. Pazy,
*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486**, DOI 10.1007/978-1-4612-5561-1 - Derek W. Robinson,
*Smooth derivations on abelian $C^\ast$-dynamical systems*, J. Austral. Math. Soc. Ser. A**42**(1987), no. 2, 247–264. MR**869749**, DOI 10.1017/S1446788700028238

## Bibliographic Information

**Fuyuan Yao**- Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
- Email: fyao@ncms1.cb.lucent.com
- Received by editor(s): January 31, 1995
- Received by editor(s) in revised form: July 19, 1995
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 1027-1032 - MSC (1991): Primary 47D05
- DOI: https://doi.org/10.1090/S0002-9939-97-03564-8
- MathSciNet review: 1346993