Isometric dilations and calculus for bounded analytic semigroups and Ritt operators
Authors:
Cédric Arhancet, Stephan Fackler and Christian Le Merdy
Journal:
Trans. Amer. Math. Soc. 369 (2017), 6899-6933
MSC (2010):
Primary 47A60; Secondary 47D06, 47A20, 22D12
DOI:
https://doi.org/10.1090/tran/6849
Published electronically:
March 17, 2017
MathSciNet review:
3683097
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Abstract | References | Similar Articles | Additional Information
Abstract: We show that any bounded analytic semigroup on (with
) whose negative generator admits a bounded
functional calculus for some
can be dilated into a bounded analytic semigroup
on a bigger
-space in such a way that
is a positive contraction for any
. We also establish a discrete analogue for Ritt operators and consider the case when
-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier's unitarization theorem.
- [ALM14] Cédric Arhancet and Christian Le Merdy, Dilation of Ritt operators on 𝐿^{𝑝}-spaces, Israel J. Math. 201 (2014), no. 1, 373–414. MR 3265289, https://doi.org/10.1007/s11856-014-1030-6
- [Are04] Wolfgang Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, Evolutionary equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, pp. 1–85. MR 2103696
- [Arh13] Cédric Arhancet, Square functions for Ritt operators on noncommutative 𝐿^{𝑝}-spaces, Math. Scand. 113 (2013), no. 2, 292–319. MR 3145184, https://doi.org/10.7146/math.scand.a-15573
- [BGM89] Earl Berkson, T. A. Gillespie, and Paul S. Muhly, Generalized analyticity in UMD spaces, Ark. Mat. 27 (1989), no. 1, 1–14. MR 1004717, https://doi.org/10.1007/BF02386355
- [Blu01a] Sönke Blunck, Analyticity and discrete maximal regularity on 𝐿_{𝑝}-spaces, J. Funct. Anal. 183 (2001), no. 1, 211–230. MR 1837537, https://doi.org/10.1006/jfan.2001.3740
- [Blu01b] Sönke Blunck, Maximal regularity of discrete and continuous time evolution equations, Studia Math. 146 (2001), no. 2, 157–176. MR 1853519, https://doi.org/10.4064/sm146-2-3
- [BR84] Charles J. K. Batty and Derek W. Robinson, Positive one-parameter semigroups on ordered Banach spaces, Acta Appl. Math. 2 (1984), no. 3-4, 221–296. MR 753696, https://doi.org/10.1007/BF02280855
- [Bur01] Donald L. Burkholder, Martingales and singular integrals in Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 233–269. MR 1863694, https://doi.org/10.1016/S1874-5849(01)80008-5
- [CDMY96] Michael Cowling, Ian Doust, Alan McIntosh, and Atsushi Yagi, Banach space operators with a bounded 𝐻^{∞} functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), no. 1, 51–89. MR 1364554
- [DF93] Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
- [Dix50] Jacques Dixmier, Les moyennes invariantes dans les semi-groupes et leurs applications, Acta Sci. Math. (Szeged) 12 (1950), 213–227 (French). MR 37470
- [DJT95] Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297
- [dLG65] K. de Leeuw and I. Glicksberg, The decomposition of certain group representations, J. Analyse Math. 15 (1965), 135–192. MR 186755, https://doi.org/10.1007/BF02787692
- [Dun11] Nick Dungey, Subordinated discrete semigroups of operators, Trans. Amer. Math. Soc. 363 (2011), no. 4, 1721–1741. MR 2746662, https://doi.org/10.1090/S0002-9947-2010-05094-9
- [Fac14] Stephan Fackler, On the structure of semigroups on 𝐿_{𝑝} with a bounded 𝐻^{∞}-calculus, Bull. Lond. Math. Soc. 46 (2014), no. 5, 1063–1076. MR 3262207, https://doi.org/10.1112/blms/bdu062
- [Fac15] Stephan Fackler, Regularity properties of sectorial operators: Extrapolation, counterexamples and generic classes, Ph.D. thesis, Universität Ulm, 2015.
- [Fig76] T. Figiel, On the moduli of convexity and smoothness, Studia Math. 56 (1976), no. 2, 121–155. MR 425581, https://doi.org/10.4064/sm-56-2-121-155
- [Fig80] T. Figiel, Uniformly convex norms on Banach lattices, Studia Math. 68 (1980), no. 3, 215–247. MR 599147, https://doi.org/10.4064/sm-68-3-215-247
- [FW06] Andreas M. Fröhlich and Lutz Weis, 𝐻^{∞} calculus and dilations, Bull. Soc. Math. France 134 (2006), no. 4, 487–508 (English, with English and French summaries). MR 2364942, https://doi.org/10.24033/bsmf.2520
- [Gol85] Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
- [Haa06] Markus Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Basel, 2006. MR 2244037
- [Hei81] S. Heinrich, Ultraproducts of 𝐿₁-predual spaces, Fund. Math. 113 (1981), no. 3, 221–234. MR 641306, https://doi.org/10.4064/fm-113-3-221-234
- [HP98] Matthias Hieber and Jan Prüss, Functional calculi for linear operators in vector-valued 𝐿^{𝑝}-spaces via the transference principle, Adv. Differential Equations 3 (1998), no. 6, 847–872. MR 1659281
- [HT10] Markus Haase and Yuri Tomilov, Domain characterizations of certain functions of power-bounded operators, Studia Math. 196 (2010), no. 3, 265–288. MR 2587299, https://doi.org/10.4064/sm196-3-4
- [Jan97] Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726
- [JLMX06] Marius Junge, Christian Le Merdy, and Quanhua Xu, 𝐻^{∞} functional calculus and square functions on noncommutative 𝐿^{𝑝}-spaces, Astérisque 305 (2006), vi+138 (English, with English and French summaries). MR 2265255
- [Kre85] Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411
- [KS01] M. A. Khamsi and B. Sims, Ultra-methods in metric fixed point theory, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 177–199. MR 1904277, https://doi.org/10.1002/9781118033074
- [KW01] N. J. Kalton and L. Weis, The 𝐻^{∞}-calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345. MR 1866491, https://doi.org/10.1007/s002080100231
- [KW04] Peer C. Kunstmann and Lutz Weis, Maximal 𝐿_{𝑝}-regularity for parabolic equations, Fourier multiplier theorems and 𝐻^{∞}-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65–311. MR 2108959, https://doi.org/10.1007/978-3-540-44653-8_2
- [KW14]
N. J. Kalton and L. Weis, The
-functional calculus and square function estimates, arXiv: http://arxiv.org/abs/1411.0472 (2014).
- [Kwa74] S. Kwapień, On Banach spaces containing 𝑐₀, Studia Math. 52 (1974), 187–188. MR 356156
- [LM98a] Christian Le Merdy, 𝐻^{∞}-functional calculus and applications to maximal regularity, Semi-groupes d’opérateurs et calcul fonctionnel (Besançon, 1998) Publ. Math. UFR Sci. Tech. Besançon, vol. 16, Univ. Franche-Comté, Besançon, 1999, pp. 41–77. MR 1768324
- [LM98b] Christian Le Merdy, The similarity problem for bounded analytic semigroups on Hilbert space, Semigroup Forum 56 (1998), no. 2, 205–224. MR 1490293, https://doi.org/10.1007/PL00005942
- [LM07] Christian Le Merdy, Square functions, bounded analytic semigroups, and applications, Perspectives in operator theory, Banach Center Publ., vol. 75, Polish Acad. Sci. Inst. Math., Warsaw, 2007, pp. 191–220. MR 2341347, https://doi.org/10.4064/bc75-0-12
- [LM10] Christian Le Merdy, 𝛾-Bounded representations of amenable groups, Adv. Math. 224 (2010), no. 4, 1641–1671. MR 2646307, https://doi.org/10.1016/j.aim.2010.01.019
- [LM14] Christian Le Merdy, 𝐻^{∞} functional calculus and square function estimates for Ritt operators, Rev. Mat. Iberoam. 30 (2014), no. 4, 1149–1190. MR 3293430, https://doi.org/10.4171/RMI/811
- [LMS01] Christian Le Merdy and Arnaud Simard, Sums of commuting operators with maximal regularity, Studia Math. 147 (2001), no. 2, 103–118. MR 1855818, https://doi.org/10.4064/sm147-2-1
- [LMX12] Christian Le Merdy and Quanhua Xu, Maximal theorems and square functions for analytic operators on 𝐿^{𝑝}-spaces, J. Lond. Math. Soc. (2) 86 (2012), no. 2, 343–365. MR 2980915, https://doi.org/10.1112/jlms/jds009
- [LT79] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
- [Lyu99] Yu. Lyubich, Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition, Studia Math. 134 (1999), no. 2, 153–167. MR 1688223, https://doi.org/10.4064/sm-134-2-153-167
- [McI86] Alan McIntosh, Operators which have an 𝐻_{∞} functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210–231. MR 912940
- [MCSA01] Celso Martínez Carracedo and Miguel Sanz Alix, The theory of fractional powers of operators, North-Holland Mathematics Studies, vol. 187, North-Holland Publishing Co., Amsterdam, 2001. MR 1850825
- [Nev93] Olavi Nevanlinna, Convergence of iterations for linear equations, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR 1217705
- [Nik02] Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. MR 1864396
- [NZ99] Béla Nagy and Jaroslav Zemánek, A resolvent condition implying power boundedness, Studia Math. 134 (1999), no. 2, 143–151. MR 1688222, https://doi.org/10.4064/sm-134-2-143-151
- [Pac69] Edward W. Packel, A semigroup analogue of Foguel’s counterexample, Proc. Amer. Math. Soc. 21 (1969), 240–244. MR 238124, https://doi.org/10.1090/S0002-9939-1969-0238124-7
- [Pat88] Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261
- [Paz83] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
- [Pis94] Gilles Pisier, Complex interpolation and regular operators between Banach lattices, Arch. Math. (Basel) 62 (1994), no. 3, 261–269. MR 1259842, https://doi.org/10.1007/BF01261367
- [Pis01] Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047
- [Pis11] Gilles Pisier, Martingales in Banach spaces (in connection with type and cotype), http://www.math.jussieu.fr/~pisier/ihp-pisier.pdf, 2011.
- [PX03] Gilles Pisier and Quanhua Xu, Non-commutative 𝐿^{𝑝}-spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517. MR 1999201, https://doi.org/10.1016/S1874-5849(03)80041-4
- [Ran01] Beata Randrianantoanina, Norm-one projections in Banach spaces, Taiwanese J. Math. 5 (2001), no. 1, 35–95. International Conference on Mathematical Analysis and its Applications (Kaohsiung, 2000). MR 1816130, https://doi.org/10.11650/twjm/1500574888
- [Sim74] Barry Simon, The 𝑃(𝜙)₂ Euclidean (quantum) field theory, Princeton University Press, Princeton, N.J., 1974. Princeton Series in Physics. MR 0489552
- [Vit05] Pascale Vitse, A band limited and Besov class functional calculus for Tadmor-Ritt operators, Arch. Math. (Basel) 85 (2005), no. 4, 374–385. MR 2174234, https://doi.org/10.1007/s00013-005-1345-7
- [vNVW13] J. M. A. M. van Neerven, M. Veraar, and L. Weis, Stochastic integration in Banach spaces - a survey, arXiv:1304.7575 (2013).
- [vNW05] J. M. A. M. van Neerven and L. Weis, Stochastic integration of functions with values in a Banach space, Studia Math. 166 (2005), no. 2, 131–170. MR 2109586, https://doi.org/10.4064/sm166-2-2
- [Wei01] Lutz Weis, Operator-valued Fourier multiplier theorems and maximal 𝐿_{𝑝}-regularity, Math. Ann. 319 (2001), no. 4, 735–758. MR 1825406, https://doi.org/10.1007/PL00004457
- [Wei06] Lutz Weis, The 𝐻^{∞} holomorphic functional calculus for sectorial operators—a survey, Partial differential equations and functional analysis, Oper. Theory Adv. Appl., vol. 168, Birkhäuser, Basel, 2006, pp. 263–294. MR 2240065, https://doi.org/10.1007/3-7643-7601-5_16
- [Yos80] Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
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Additional Information
Cédric Arhancet
Affiliation:
Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France
Email:
cedric.arhancet@univ-fcomte.fr
Stephan Fackler
Affiliation:
Institute of Applied Analysis, University of Ulm, Helmholtzstrasse 18, 89069 Ulm, Germany
Email:
stephan.fackler@uni-ulm.de
Christian Le Merdy
Affiliation:
Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France
Email:
christian.lemerdy@univ-fcomte.fr
DOI:
https://doi.org/10.1090/tran/6849
Keywords:
Dilation,
Ritt operator,
sectorial operator,
group representation,
functional calculus,
semigroup,
amenable group
Received by editor(s):
April 28, 2015
Received by editor(s) in revised form:
September 28, 2015
Published electronically:
March 17, 2017
Article copyright:
© Copyright 2017
American Mathematical Society