Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity

Colombo, Fabrizio; Gantner, Jonathan

doi:10.1090/tran/7013

Fractional powers of quaternionic operators and Kato’s formula using slice hyperholomorphicity
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by Fabrizio Colombo and Jonathan Gantner PDF

Trans. Amer. Math. Soc. 370 (2018), 1045-1100 Request permission

Abstract:

In this paper we introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice hyperholomorphic functions and on the $S$-resolvent operators of the quaternionic functional calculus. The integral representation formulas of the fractional powers and the quaternionic version of Kato’s formula are based on the notion of $S$-spectrum of a quaternionic operator.

The proofs of several properties of the fractional powers of quaternionic operators rely on the $S$-resolvent equation. This equation, which is very important and of independent interest, has already been introduced in the case of bounded quaternionic operators, but for the case of unbounded operators some additional considerations have to be taken into account. Moreover, we introduce a new series expansion for the pseudo-resolvent, which is of independent interest and allows to investigate the behavior of the $S$-resolvents close to the $S$-spectrum.

The paper is addressed to researchers working in operator theory and in complex analysis.

References

Stephen L. Adler, Quaternionic quantum mechanics and quantum fields, International Series of Monographs on Physics, vol. 88, The Clarendon Press, Oxford University Press, New York, 1995. MR 1333599
Daniel Alpay, The Schur algorithm, reproducing kernel spaces and system theory, SMF/AMS Texts and Monographs, vol. 5, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1998 French original by Stephen S. Wilson. MR 1839648
Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, and David P. Kimsey, Functions of the infinitesimal generator of a strongly continuous quaternionic group, Anal. Appl. (Singap.) 15 (2017), no. 2, 279–311. MR 3600960, DOI 10.1142/S021953051650007X
Daniel Alpay, Fabrizio Colombo, Jonathan Gantner, and Irene Sabadini, A new resolvent equation for the $S$-functional calculus, J. Geom. Anal. 25 (2015), no. 3, 1939–1968. MR 3358079, DOI 10.1007/s12220-014-9499-9
Daniel Alpay, Fabrizio Colombo, Izchak Lewkowicz, and Irene Sabadini, Realizations of slice hyperholomorphic generalized contractive and positive functions, Milan J. Math. 83 (2015), no. 1, 91–144. MR 3347393, DOI 10.1007/s00032-014-0231-9
Daniel Alpay, Fabrizio Colombo, and David P. Kimsey, The spectral theorem for quaternionic unbounded normal operators based on the $S$-spectrum, J. Math. Phys. 57 (2016), no. 2, 023503, 27. MR 3450567, DOI 10.1063/1.4940051
Daniel Alpay, Fabrizio Colombo, David P. Kimsey, and Irene Sabadini, The spectral theorem for unitary operators based on the $S$-spectrum, Milan J. Math. 84 (2016), no. 1, 41–61. MR 3503195, DOI 10.1007/s00032-015-0249-7
Daniel Alpay, Fabrizio Colombo, and Irene Sabadini, Krein-Langer factorization and related topics in the slice hyperholomorphic setting, J. Geom. Anal. 24 (2014), no. 2, 843–872. MR 3192300, DOI 10.1007/s12220-012-9358-5
Daniel Alpay, Fabrizio Colombo, and Irene Sabadini, Perturbation of the generator of a quaternionic evolution operator, Anal. Appl. (Singap.) 13 (2015), no. 4, 347–370. MR 3340665, DOI 10.1142/S0219530514500249
Daniel Alpay, Fabrizio Colombo, and Irene Sabadini, Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions, J. Anal. Math. 121 (2013), 87–125. MR 3127378, DOI 10.1007/s11854-013-0028-8
Daniel Alpay, Fabrizio Colombo, and Irene Sabadini, Schur functions and their realizations in the slice hyperholomorphic setting, Integral Equations Operator Theory 72 (2012), no. 2, 253–289. MR 2872478, DOI 10.1007/s00020-011-1935-7
Daniel Alpay, Fabrizio Colombo, and Irene Sabadini, Slice hyperholomorphic Schur analysis, Operator Theory: Advances and Applications, vol. 256, Birkhäuser/Springer, Cham, 2016. MR 3585855, DOI 10.1007/978-3-319-42514-6
Daniel Alpay, Aad Dijksma, James Rovnyak, and Hendrik de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Operator Theory: Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997. MR 1465432, DOI 10.1007/978-3-0348-8908-7
A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419–437. MR 115096
Garrett Birkhoff and John von Neumann, The logic of quantum mechanics, Ann. of Math. (2) 37 (1936), no. 4, 823–843. MR 1503312, DOI 10.2307/1968621
Fabrizio Colombo and Jonathan Gantner, On power series expansions of the $S$-resolvent operator and the Taylor formula, J. Geom. Phys. 110 (2016), 154–175. MR 3566106, DOI 10.1016/j.geomphys.2016.07.018
Fabrizio Colombo and Irene Sabadini, On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 (2009), no. 3, 601–627. MR 2496568, DOI 10.1007/s12220-009-9075-x
Fabrizio Colombo and Irene Sabadini, On the formulations of the quaternionic functional calculus, J. Geom. Phys. 60 (2010), no. 10, 1490–1508. MR 2661152, DOI 10.1016/j.geomphys.2010.05.014
Fabrizio Colombo and Irene Sabadini, The quaternionic evolution operator, Adv. Math. 227 (2011), no. 5, 1772–1805. MR 2803786, DOI 10.1016/j.aim.2011.04.001
Fabrizio Colombo and Irene Sabadini, On some notions of spectra for quaternionic operators and for $n$-tuples of operators, C. R. Math. Acad. Sci. Paris 350 (2012), no. 7-8, 399–402 (English, with English and French summaries). MR 2922091, DOI 10.1016/j.crma.2012.03.017
Fabrizio Colombo and Irene Sabadini, Some remarks on the $\scr S$-spectrum, Complex Var. Elliptic Equ. 58 (2013), no. 1, 1–6. MR 3011972, DOI 10.1080/17476933.2010.551191
Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa, Noncommutative functional calculus, Progress in Mathematics, vol. 289, Birkhäuser/Springer Basel AG, Basel, 2011. Theory and applications of slice hyperholomorphic functions. MR 2752913, DOI 10.1007/978-3-0348-0110-2
N. Dunford, J. Schwartz, Linear Operators, part I: General Theory, J. Wiley and Sons, New York, 1988.
N. Dunford, J. Schwartz, Linear Operators, part II: Spectral theory, J. Wiley and Sons, New York, 1988.
Gérard Emch, Mécanique quantique quaternionienne et relativité restreinte. I, Helv. Phys. Acta 36 (1963), 739–769 (French, with English summary). MR 176811
K. J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.
Douglas R. Farenick and Barbara A. F. Pidkowich, The spectral theorem in quaternions, Linear Algebra Appl. 371 (2003), 75–102. MR 1997364, DOI 10.1016/S0024-3795(03)00420-8
David Finkelstein, Josef M. Jauch, Samuel Schiminovich, and David Speiser, Foundations of quaternion quantum mechanics, J. Mathematical Phys. 3 (1962), 207–220. MR 137500, DOI 10.1063/1.1703794
J. Gantner, Slice hyperholomorphic functions and the quaternionic functional calculus, Master Thesis, 2014, Vienna University of Technology, http://www.asc.tuwien.ac.at/~funkana/downloads_general/dipl_gantner.pdf
Jonathan Gantner, A direct approach to the $S$-functional calculus for closed operators, J. Operator Theory 77 (2017), no. 2, 287–331. MR 3634508, DOI 10.7900/jot.2017mar24.2092
Graziano Gentili, Caterina Stoppato, and Daniele C. Struppa, Regular functions of a quaternionic variable, Springer Monographs in Mathematics, Springer, Heidelberg, 2013. MR 3013643, DOI 10.1007/978-3-642-33871-7
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, DOI 10.1142/S0129055X13500062
R. Ghiloni, V. Moretti, A. Perotti, Spectral properties of compact normal quaternionic operators, in: S. Bernstein, U. Kähler, I. Sabadini, F. Sommen (eds.): Hypercomplex Analysis: New Perspectives and Applications, Trends in Mathematics, Birkhäuser, Basel, (2014), 133–143.
R. Ghiloni and A. Perotti, Slice regular functions on real alternative algebras, Adv. Math. 226 (2011), no. 2, 1662–1691. MR 2737796, DOI 10.1016/j.aim.2010.08.015
Riccardo Ghiloni and Vincenzo Recupero, Semigroups over real alternative *-algebras: generation theorems and spherical sectorial operators, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2645–2678. MR 3449252, DOI 10.1090/tran/6399
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
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
Alberto Guzman, Fractional-power semigroups of growth $n$, J. Functional Analysis 30 (1978), no. 2, 223–237. MR 515226, DOI 10.1016/0022-1236(78)90071-X
Alberto Guzman, Growth properties of semigroups generated by fractional powers of certain linear operators, J. Functional Analysis 23 (1976), no. 4, 331–352. MR 0428104, DOI 10.1016/0022-1236(76)90060-4
Alberto Guzman, Further study of growth of fractional-power semigroups, J. Functional Analysis 29 (1978), no. 2, 133–141. MR 504454, DOI 10.1016/0022-1236(78)90001-0
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
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, DOI 10.1016/0003-4916(84)90068-X
Shmuel Kantorovitz, Topics in operator semigroups, Progress in Mathematics, vol. 281, Birkhäuser Boston, Ltd., Boston, MA, 2010. MR 2574324, DOI 10.1007/978-0-8176-4932-6
Tosio Kato, Note on fractional powers of linear operators, Proc. Japan Acad. 36 (1960), 94–96. MR 121666
Hikosaburo Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285–346. MR 201985
Hikosaburo Komatsu, Fractional powers of operators. II. Interpolation spaces, Pacific J. Math. 21 (1967), 89–111. MR 206716
Hikosaburo Komatsu, Fractional powers of operators. III. Negative powers, J. Math. Soc. Japan 21 (1969), 205–220. MR 242005, DOI 10.2969/jmsj/02120205
Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR 1329547, DOI 10.1007/978-3-0348-9234-6
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
C. S. Sharma and T. J. Coulson, Spectral theory for unitary operators on a quaternionic Hilbert space, J. Math. Phys. 28 (1987), no. 9, 1941–1946. MR 904408, DOI 10.1063/1.527458
G. A. G. Soukhomlinoff, Über Fortsetzung von linearen Funktionalen in linearen komplexen Räumen und linearen Quaternionräumen, Rec. Math. [Mat. Sbornik] N.S., 3(45)-2(1938), 353–358
K. Viswanath, Normal operations on quaternionic Hilbert spaces, Trans. Amer. Math. Soc. 162 (1971), 337–350. MR 284843, DOI 10.1090/S0002-9947-1971-0284843-X
Jiro Watanabe, On some properties of fractional powers of linear operators, Proc. Japan Acad. 37 (1961), 273–275. MR 140958
Kôsaku Yosida, Fractional powers of infinitesimal generators and the analyticity of the semi-groups generated by them, Proc. Japan Acad. 36 (1960), 86–89. MR 121665