Abstract
Optimal upper bounds are given for the norm of the semigroup (e −tV)t≥0, where V is the classical Volterra operator acting on L p [0,1], 1≤p≤∞. In particular, for every p∈[1,∞] we prove that
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bermudo, S., Montes-Rodríguez, A., Shkarin, S.: Orbits of operators commuting with the Volterra operator. J. Math. Pures Appl. 89, 145–173 (2008)
Lyubich, Y.: The power boundedness and resolvent conditions for functions of the classical Volterra operator. Stud. Math. 196, 41–63 (2010)
Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford University Press, New York (1985)
Montes-Rodríguez, A., Sánchez-Álvarez, J., Zemánek, J.: Uniform Abel-Kreiss boundedness and the extremal behaviour of the Volterra operator. Proc. Lond. Math. Soc. 91, 761–788 (2005)
Hille, E.: Remarks on ergodic theorems. Trans. Am. Math. Soc. 57, 246–269 (1945)
Erde’lyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)
Katznelson, Y.: An Introduction to Harmonic Analysis. Dover, New York (1976)
Zwart, H.: Is A −1 an infinitesimal generator. In: Perspectives in Operator Theory, Warsaw, 2004. Banach Center Publications, vol. 75, pp. 303–313. Polish Academy of Sciences, Institute of Mathematics, Warsaw (2007)
Gomilko, A.M., Zwart, H., Tomilov, Y.: Inverse operator of the generator of a C 0-semigroup. Mat. Sb. 198, 35–50 (2007) [in Russian]; English transl. in Sb. Math. 198, 1095–1110 (2007)
deLaubenfels, R.: Inverses of generators. Proc. Am. Math. Soc. 104, 443–448 (1988)
Komatsu, H.: Fractional powers of operators. Pac. J. Math. 19, 285–346 (1966)
deLaubenfels, R.: Inverses of generators of nonanalytic semigroups. Stud. Math. 191, 11–38 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jerome A. Goldstein.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Gomilko, O. Optimal estimates for the semigroup generated by the classical Volterra operator on L p -spaces. Semigroup Forum 83, 343–350 (2011). https://doi.org/10.1007/s00233-011-9317-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-011-9317-7