RETRACTED: Transference Principles for the Series of Semigroups with a Theorem of Peller

Simon Joseph^1*, Ahmed Sufyan², Hala Taha³, Ranya Tahir⁴

¹Department of Mathematics, College of Education, Dr. John Garang Memorial University of Science and Technology, Bor, South Sudan.
²Department of Mathematics, Ministry of Education, Muscat, Sultanate of Oman.
³Department of Mathematics, College of Science, Princess Nourah bint Abdulrahman University, Al-Riyadh, KSA.
⁴Department of Mathematics, Faculty of Science, Jazan University, Jazan, KSA.

DOI: 10.4236/apm.2019.92009                

Abstract

Short Retraction Notice

The paper does not meet the standards of "Advances in Pure Mathematics". This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction Guidelines. The aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused.

Editor guiding this retraction: Prof. Leo Depuydt (EIC of APM). Please see the article page for more details.

The full retraction notice in PDF is preceding the original paper which is marked "RETRACTED".

Keywords

Transference, Operator Semigroup, Functional Calculus, Analytic Besov, Peller, γ-Boundedness, γ-Radonifying, γ-Summing, Power-Bounded Operator

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Haase, M. (2011) Transference Principles for Semigroups and a Theorem of Peller. Journal of Functional Analysis, 261, 2959-2998. https://doi.org/10.1016/j.jfa.2011.07.019
[2]	Calderon, A.P. (1968) Ergodic Theory and Translation-Invariant Operators. Proceedings of the National Academy of Sciences of the United States of America, 59, 349-353. https://doi.org/10.1073/pnas.59.2.349
[3]	Coiffman, R.R. and Weiss, G. (1976) Transference Methods in Analysis. CBMS Regional Conference Series in Mathematics, vol. 31. American Mathematical Society, Providence, RI.
[4]	Coiffman, R. and Weiss, G. (1977) Some Examples of Transference Methods in Harmonic Analysis. Academic Press, London, 33-45. https://doi.org/10.1090/cbms/031
[5]	Arendt, W., Batty, C.J.K., Hieber, M. and Neubrander, F. (1989) Generalized Analyticity in UMD Space. Arkiv for Matematik, 27, 1-14. https://doi.org/10.1007/BF02386355
[6]	Blower, G. (2000) Maximal Functions and Transference for Groups of Operators. Proceedings of the Edinburgh Mathematical Society, 43, 57-71. https://doi.org/10.1017/S0013091500020691
[7]	Haase, M. (2007) Functional Calculus for Groups and Applications to Evolution Equations. Journal of Evolution Equations, 11, 529-554. https://doi.org/10.1007/s00028-007-0313-z
[8]	Haase, M. (2009) A Transference Principle for General Groups and Functional Calculus on UMD Space. Mathematische Zeitschrift, 262, 281-299. https://doi.org/10.1007/s00209-008-0373-y
[9]	von Neumann, J. (1951) Eine Spektraltheoorie fur allgemeine: Operator eneinesunitaren Raumes. Mathematische Nachrichten, 4, 258-281.
[10]	Lebow, A. (1968) A Power-Bounded Operator That Is Not Polynomially Bounded. Michigan Mathematical Journal, 15, 397-399. https://doi.org/10.1307/mmj/1029000094
[11]	Arendt, W., Batty, C.J.K., Hieber, M. and Neubrander, F. (2001) Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics. vol. 96, Birkhauser, Basel, 523 p. https://doi.org/10.1007/978-3-0348-5075-9
[12]	Engel, K.-J. and Nagel, R. (2000) One-Parameter Semigroups for Linear Evolution Equations, vol. 194. Springer, Berlin/Heidelberg.
[13]	Haase, M. (2006) The Functional Calculus for Sectorial Operator. Operator Theory: Advances and Applications. Birkhauser, Basel. https://doi.org/10.1007/3-7643-7698-8
[14]	Le Merdy, C. (2000) A Bounded Compact Semigroup on Hilbert Space Not Similar to a Contaction One. Birkhauser, Basel, 213-216.
[15]	Eisner, T. and Zwart, H. (2008) The Growth of a C0-Semigroup Characterized by Its Cogenerator. Journal of Evolution Equations, 8, 749-764. https://doi.org/10.1007/s00028-008-0416-1
[16]	Peller, V.V. (1982) Estimates of Functions of Power Bounded Operators on Hilbert Spaces. Journal of Operator Theory, 7, 341-372.
[17]	Vitse, P. (2005) A Band Limited and Besov Class Functional Calculus for Tadmor-Rittoperators. Archiv der Mathematik (Basel), 85, 374-385. https://doi.org/10.1007/s00013-005-1345-7
[18]	Vitse, P. (2005) A Besov Class Functional Calculus for Bounded Holomorphic Semigroups. Journal of Functional Analysis, 228, 245-269. https://doi.org/10.1016/j.jfa.2005.01.010
[19]	Kalton, N. and Weis, L. (2004) The H∞-Functional Calculus and Square Function Estimate. Unpublished Manuscript.
[20]	Le Merdy, C. (2010) γ-Bounded Representations of Amenable Groups. Advances in Mathematics, 224, 1641-1671. https://doi.org/10.1016/j.aim.2010.01.019
[21]	Burkholder, D.L. (2001) Martingales and Singular Integrals in Banach Spaces. In: Handbook of the Geometry of Banach Spaces, vol. 1, North-Holland, Amsterdam, 233-269. https://doi.org/10.1016/S1874-5849(01)80008-5
[22]	Hieber, M. and Pruss, J. (1998) Functional Calculi for Linear Operators in Vector-Valued Lp-Spaces via the Transference Principle. Advances in Difference Equations, 3, 847-872.
[23]	Boyadzhiev, K. and Delaubenfels, R. (1994) Spectral Theorem for Unbounded Strongly Continuous Groups on a Hilbert Space. Proceedings of the American Mathematical Society, 120, 127-136. https://doi.org/10.1090/S0002-9939-1994-1186983-0
[24]	Mclntosh, A. (1986) Operator Which Have an H∞ Functional Calculus. University of Canberra, Bruce, 210-231.
[25]	Haase, M. (2009) The Group Reduction for Bounded Cosine Functions on UMD Spaces. Mathematische Zeitschrif, 262, 281-299. https://doi.org/10.1007/s00209-008-0373-y
[26]	Hille, E. and Phillips, R.S. (1974) Functional Analysis and Semi Group. The American Mathematical Society, Providence, 808 p.
[27]	van Neerven, J. (2010) γ-Radoniifying Operators—A Survey. Proceedings of the CMA, 44, 1-62.
[28]	Haase, M. (2006) Semigroup Theory via Functional Calculus. Preprint.
[29]	Garling, D.J.H. (2007) Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511755217