Banach spaces and their applications in analysis / Espaces de Banach et applications à l'analyse

Collection Banach spaces and their applications in analysis / Espaces de Banach et applications à l'analyse

Organizer(s) Albiac, Fernando ; Casazza, Peter G. ; Godefroy, Gilles
Date(s) 12/01/2015 - 16/01/2015
00:00:00 / 00:00:00
2 5

Perturbations of the holomorphic functional calculus: differential operators versus general sectorial operators

By Pierre Portal

Nigel Kalton played a prominent role in the development of a holomorphic functional calculus for unbounded sectorial operators. He showed, in particular, that such a calculus is highly unstable under perturbation: given an operator $D$ with a bounded functional calculus, fairly stringent conditions have to be imposed on a perturbation $B$ for $DB$ to also have a bounded functional calculus. Nigel, however, often mentioned that, while these results give a fairly complete picture of what is true at a pure operator theoretic level, more should be true for special classes of differential operators. In this talk, I will briefly review Nigel's general results before focusing on differential operators with perturbed coefficients acting on $L_p(\mathbb{R}^{n})$. I will present, in particular, recent joint work with $D$. Frey and A. McIntosh that demonstrates how stable the functional calculus is in this case. The emphasis will be on trying, as suggested by Nigel, to understand what makes differential operators so special from an operator theoretic point of view.

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.18665003
  • Cite this video Portal, Pierre (13/01/2015). Perturbations of the holomorphic functional calculus: differential operators versus general sectorial operators. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18665003
  • URL https://dx.doi.org/10.24350/CIRM.V.18665003

Bibliography

  • Axelsson, A., Keith, S., & McIntosh, A. (2006). Quadratic estimates and functional calculi of perturbed Dirac operators. Inventiones Mathematicae, 163(3), 455-497 - http://dx.doi.org/10.1007/s00222-005-0464-x
  • Carleson, L. (1962). Interpolation of bounded analytic functions and the corona problem. Annals of Mathematics. Second Series, 76, 547-599 - http://dx.doi.org/10.2307/1970375
  • Cohn, W., & Verbitsky, I. (2000). Factorization of tent spaces and Hankel operators. Journal of Functional Analysis, 175(2), 308-329 - http://dx.doi.org/10.1006/jfan.2000.3589
  • Coifman, R.R., Meyer, Y., & Stein, E.M. (1985). Some new function spaces and their applications to harmonic analysis. Journal of Functional Analysis, 62(2), 304-335 - http://dx.doi.org/10.1016/0022-1236(85)90007-2
  • Cowling, M., Doust, I., McIntosh, A., & Yagi, A. (1996). Banach space operators with a bounded $H^{\infty}$ functional calculus. Journal of the Australian Mathematical Society (Series A), 60(1), 51-89 - http://dx.doi.org/10.1017/s1446788700037393
  • Frey, D., McIntosh, A., & Portal, P. (2014). Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in $L^p$. <arXiv:1407.4774> - http://arxiv.org/abs/1407.4774v2
  • Hytönen, T., & McIntosch, A. (2010). Stability in p of the $H^{\infty}$-calculus of first-order systems in $L^p$. In A. Hassell, A. McIntosh, & R. Taggart (Eds.), The AMSI-ANU workshop on spectral theory and harmonic analysis. Proceedings of the workshop, Canberra, Australia, July 13–17, 2009 (pp. 167-181). Canberra: Australian National University. (Proceedings of the Centre for Mathematics and its Applications, 44) - https://www.zbmath.org/?q=an:1252.47014
  • Hytönen, T., McIntosh, A., & Portal, P. (2008). Kato's square root problem in Banach spaces. Journal of Functional Analysis, 254(3), 675-726 - http://dx.doi.org/10.1016/j.jfa.2007.10.006
  • Kalton, N.J. (2007). Perturbations of the $H^{\infty}$-calculus. Collectanea Mathematica, 58(3), 291-325 - https://eudml.org/doc/42036

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