Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Semigroups over real alternative *-algebras: Generation theorems and spherical sectorial operators


Authors: Riccardo Ghiloni and Vincenzo Recupero
Journal: Trans. Amer. Math. Soc. 368 (2016), 2645-2678
MSC (2010): Primary 30G35, 47D03, 47A60, 47A10
DOI: https://doi.org/10.1090/tran/6399
Published electronically: April 14, 2015
MathSciNet review: 3449252
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is twofold. On one hand, generalizing some recent results obtained in the quaternionic setting, but using simpler techniques, we prove the generation theorems for semigroups in Banach spaces whose set of scalars belongs to the class of real alternative *-algebras, which includes, besides real and complex numbers, quaternions, octonions and Clifford algebras. On the other hand, in this new general framework, we introduce the notion of spherical sectorial operator and we prove that a spherical sectorial operator generates a semigroup that can be represented by a Cauchy integral formula. It follows that such a semigroup is analytic in time.


References [Enhancements On Off] (What's this?)

  • [1] S. Adler, Quaternionic Quantum Field Theory, Oxford University Press, 1995.
  • [2] Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, Vol. 13. MR 0417223 (54 #5281)
  • [3] Garrett Birkhoff and John von Neumann, The logic of quantum mechanics, Ann. of Math. (2) 37 (1936), no. 4, 823-843. MR 1503312, https://doi.org/10.2307/1968621
  • [4] Fabrizio Colombo, Graziano Gentili, and Irene Sabadini, A Cauchy kernel for slice regular functions, Ann. Global Anal. Geom. 37 (2010), no. 4, 361-378. MR 2601496 (2011b:30118), https://doi.org/10.1007/s10455-009-9191-7
  • [5] Fabrizio Colombo, Graziano Gentili, Irene Sabadini, and Daniele C. Struppa, Non commutative functional calculus: bounded operators, Complex Anal. Oper. Theory 4 (2010), no. 4, 821-843. MR 2735309 (2011k:47025), https://doi.org/10.1007/s11785-009-0015-3
  • [6] Fabrizio Colombo, Graziano Gentili, Irene Sabadini, and Daniele C. Struppa, Non-commutative functional calculus: unbounded operators, J. Geom. Phys. 60 (2010), no. 2, 251-259. MR 2587392 (2011h:47028), https://doi.org/10.1016/j.geomphys.2009.09.011
  • [7] Fabrizio Colombo and Irene Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 (2009), no. 3, 601-627. MR 2496568 (2010h:47024), https://doi.org/10.1007/s12220-009-9075-x
  • [8] Fabrizio Colombo and Irene Sabadini, On the formulations of the quaternionic functional calculus, J. Geom. Phys. 60 (2010), no. 10, 1490-1508. MR 2661152 (2011g:47031), https://doi.org/10.1016/j.geomphys.2010.05.014
  • [9] Fabrizio Colombo and Irene Sabadini, The quaternionic evolution operator, Adv. Math. 227 (2011), no. 5, 1772-1805. MR 2803786 (2012g:47014), https://doi.org/10.1016/j.aim.2011.04.001
  • [10] Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254 (2008), no. 8, 2255-2274. MR 2402108 (2009e:47017), https://doi.org/10.1016/j.jfa.2007.12.008
  • [11] Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009), 385-403. MR 2520116 (2010e:30039), https://doi.org/10.1007/s11856-009-0055-4
  • [12] Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa, Noncommutative functional calculus, Theory and applications of slice hyperholomorphic functions, Progress in Mathematics, vol. 289, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2752913
  • [13] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162 (90g:47001a)
  • [14] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert, Numbers, Graduate Texts in Mathematics, vol. 123, Springer-Verlag, New York, 1991. With an introduction by K. Lamotke; Translated from the second 1988 German edition by H. L. S. Orde; Translation edited and with a preface by J. H. Ewing; Readings in Mathematics. MR 1415833 (97f:00001)
  • [15] Gérard Emch, Mécanique quantique quaternionienne et relativité restreinte. I, Helv. Phys. Acta 36 (1963), 739-769 (French, with English summary). MR 0176811 (31 #1083)
  • [16] Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989 (2000i:47075)
  • [17] David Finkelstein, Josef M. Jauch, Samuel Schiminovich, and David Speiser, Foundations of quaternion quantum mechanics, J. Mathematical Phys. 3 (1962), 207-220. MR 0137500 (25 #952)
  • [18] Graziano Gentili and Daniele C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), no. 1, 279-301. MR 2353257 (2008h:30052), https://doi.org/10.1016/j.aim.2007.05.010
  • [19] Graziano Gentili and Daniele C. Struppa, Regular functions on the space of Cayley numbers, Rocky Mountain J. Math. 40 (2010), no. 1, 225-241. MR 2607115 (2011c:30124), https://doi.org/10.1216/RMJ-2010-40-1-225
  • [20] Riccardo Ghiloni, Valter Moretti, and Alessandro Perotti, Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013), no. 4, 1350006, 83. MR 3062919, https://doi.org/10.1142/S0129055X13500062
  • [21] R. Ghiloni and A. Perotti, Slice regular functions on real alternative algebras, Adv. Math. 226 (2011), no. 2, 1662-1691. MR 2737796 (2012e:30061), https://doi.org/10.1016/j.aim.2010.08.015
  • [22] Riccardo Ghiloni and Alessandro Perotti, Volume Cauchy formulas for slice functions on real associative *-algebras, Complex Var. Elliptic Equ. 58 (2013), no. 12, 1701-1714. MR 3170730, https://doi.org/10.1080/17476933.2012.709851
  • [23] Riccardo Ghiloni and Alessandro Perotti, Global differential equations for slice regular functions, Math. Nachr. 287 (2014), no. 5-6, 561-573. MR 3193936, https://doi.org/10.1002/mana.201200318
  • [24] H. H. Goldstine and L. P. Horwitz, Hilbert space with non-associative scalars. I, Math. Ann. 154 (1964), 1-27. MR 0161122 (28 #4331)
  • [25] Klaus Gürlebeck, Klaus Habetha, and Wolfgang Sprößig, Holomorphic functions in the plane and $ n$-dimensional space, Birkhäuser Verlag, Basel, 2008. Translated from the 2006 German original; With 1 CD-ROM (Windows and UNIX). MR 2369875 (2009a:30102)
  • [26] L. P. Horwitz and L. C. Biedenharn, Quaternion quantum mechanics: second quantization and gauge fields, Ann. Physics 157 (1984), no. 2, 432-488. MR 768240 (86h:81053a), https://doi.org/10.1016/0003-4916(84)90068-X
  • [27] Irving Kaplansky, Normed algebras, Duke Math. J. 16 (1949), 399-418. MR 0031193 (11,115d)
  • [28] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486 (85g:47061)
  • [29] Richard D. Schafer, An introduction to nonassociative algebras, Dover Publications, Inc., New York, 1995. Corrected reprint of the 1966 original. MR 1375235 (96j:17001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30G35, 47D03, 47A60, 47A10

Retrieve articles in all journals with MSC (2010): 30G35, 47D03, 47A60, 47A10


Additional Information

Riccardo Ghiloni
Affiliation: Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123 Povo-Trento (TN), Italy
Email: ghiloni@science.unitn.it

Vincenzo Recupero
Affiliation: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli, Abruzzi 24, 10129 Torino, Italy
Email: vincenzo.recupero@polito.it

DOI: https://doi.org/10.1090/tran/6399
Keywords: Functions of hypercomplex variables, functional calculus, semigroups of linear operators, spectrum, resolvent
Received by editor(s): December 3, 2013
Received by editor(s) in revised form: January 26, 2014
Published electronically: April 14, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society