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Functions of the infinitesimal generator of a strongly continuous quaternionic group

    The quaternionic analogue of the Riesz–Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that T is the quaternionic infinitesimal generator of a strongly continuous group of operators (𝒵T(t))t and we show how we can define bounded operators f(T), where f belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace–Stieltjes transform. This class includes functions that are slice regular on the S-spectrum of T but not necessarily at infinity. Moreover, we establish the relation between f(T) and the quaternionic functional calculus and we study the problem of finding the inverse of f(T).

    AMSC: 47A10, 47A60

    References

    • 1. S. Adler , Quaternionic Quantum Field Theory (Oxford University Press, 1995). Google Scholar
    • 2. S. Agrawal and S. H. Kulkarni , An analogue of the Riesz-representation theorem, Novi Sad J. Math. 30 (2000) 143–154. Google Scholar
    • 3. D. Alpay, The Schur Algorithm, Reproducing Kernel Spaces and System Theory (American Mathematical Society, Providence, RI, 2001); Translated from the 1998 French original by Stephen S. Wilson, Panoramas et Synthèses. Google Scholar
    • 4. D. Alpay, F. Colombo, J. Gantner and I. Sabadini , A new resolvent equation for the S-functional calculus, J. Geom. Anal. 25 (2015) 1939–1968. Crossref, ISIGoogle Scholar
    • 5. D. Alpay, F. Colombo and D. P. Kimsey , The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum, J. Math. Phys. 57(2) (2016) 023503, 27 pp. Crossref, ISIGoogle Scholar
    • 6. D. Alpay, F. Colombo, D. P. Kimsey and I. Sabadini, The spectral theorem for unitary operators based on the S-spectrum, to appear in Milan J. Math., doi: 10.1007/s00032-015-0249-7. Google Scholar
    • 7. D. Alpay, F. Colombo and I. Sabadini , Schur functions and their realizations in the slice hyperholomorphic setting, Integral Equations Operator Theory 72 (2012) 253–289. Crossref, ISIGoogle Scholar
    • 8. D. Alpay, F. Colombo and I. Sabadini , Pontryagin–de Branges–Rovnyak spaces of slice hyperholomorphic functions, J. Anal. Math. 121 (2013) 87–125. Crossref, ISIGoogle Scholar
    • 9. D. Alpay, F. Colombo and I. Sabadini , Krein–Langer factorization and related topics in the slice hyperholomorphic setting, J. Geom. Anal. 24 (2014) 843–872. Crossref, ISIGoogle Scholar
    • 10. D. Alpay, F. Colombo and I. Sabadini , Perturbation of the generator of a quaternionic evolution operator, Anal. Appl. (Singap.) 13 (2015) 347–370. Link, ISIGoogle Scholar
    • 11. D. Alpay, A. Dijksma, J. Rovnyak and H. de Snoo , Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces, Operator Theory: Advances and Applications, Vol. 96 (Birkhäuser, Basel, 1997). CrossrefGoogle Scholar
    • 12. G. Birkhoff and J. von Neumann , The logic of quantum mechanics, Ann. of Math. 37 (1936) 823–843. CrossrefGoogle Scholar
    • 13. F. Colombo and I. Sabadini , On some properties of the quaternionic functional calculus, J. Geom. Anal. 19 (2009) 601–627. Crossref, ISIGoogle Scholar
    • 14. F. Colombo and I. Sabadini , On the formulations of the quaternionic functional calculus, J. Geom. Phys. 60 (2010) 1490–1508. Crossref, ISIGoogle Scholar
    • 15. F. Colombo and I. Sabadini , The quaternionic evolution operator, Adv. Math., 227 (2011) 1772–1805. Crossref, ISIGoogle Scholar
    • 16. F. Colombo, I. Sabadini and D. C. Struppa , Noncommutative Functional Calculus. Theory and Applications of Slice Regular Functions, Progress in Mathematics, Vol. 289 (Birkhäuser/Springer Basel AG, Basel, 2011). CrossrefGoogle Scholar
    • 17. N. Dunford and J. Schwartz , Linear Operators: Part I: General Theory (Wiley, 1988). Google Scholar
    • 18. N. Dunford and J. Schwartz , Linear Operators: Part II: Spectral Theory (Wiley, 1988). Google Scholar
    • 19. G. Emch , Mécanique quantique quaternionienne et relativité restreinte, I, Helv. Phys. Acta 36 (1963) 739–769. Google Scholar
    • 20. K. J. Engel and R. Nagel , A Short Course on Operator Semigroups, Universitext (Springer, New York, 2006). Google Scholar
    • 21. D. R. Farenick and B. A. F. Pidkowich , The spectral theorem in quaternions, Linear Algebra Appl. 371 (2003) 75–102. Crossref, ISIGoogle Scholar
    • 22. D. Finkelstein, J. M. Jauch, S. Schiminovich and D. Speiser , Foundations of quaternion quantum mechanics, J. Math. Phys. 3 (1962) 207–220. Crossref, ISIGoogle Scholar
    • 23. G. Gentili, C. Stoppato and D. C. Struppa , Regular Functions of a Quaternionic Variable, Springer Monographs in Mathematics (Springer, Heidelberg, 2013). CrossrefGoogle Scholar
    • 24. R. Ghiloni, V. Moretti and A. Perotti , Continuous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys. 25 (2013) 1350006, 83 pp. Link, ISIGoogle Scholar
    • 25. R. Ghiloni, V. Moretti and A. Perotti , Spectral properties of compact normal quaternionic operators, in Hypercomplex Analysis: New Perspectives and Applications, Trends in Mathematics (Springer, 2014), pp. 133–143. CrossrefGoogle Scholar
    • 26. R. Ghiloni and A. Perotti , Slice regular functions on real alternative algebras, Adv. Math. 226 (2011) 1662–1691. Crossref, ISIGoogle Scholar
    • 27. R. Ghiloni and V. Recupero , Semigroups over real alternative *-algebras: Generation theorems and spherical sectorial operators, Trans. Amer. Math. Soc. 368(4) (2016) 2645–2678. Crossref, ISIGoogle Scholar
    • 28. E. Hille and R. S. Phillips , Functional Analysis and Semi-groups, rev. edn., American Mathematical Society Colloquium Publications, Vol. 31 (American Mathematical Society, Providence, RI, 1957). Google Scholar
    • 29. L. P. Horwitz and L. C. Biedenharn , Quaternion quantum mechanics: Second quantization and gauge fields, Ann. Phys. 157 (1984) 432–488. Crossref, ISIGoogle Scholar
    • 30. S. Kantorovitz , Topics in Operator Semigroups, Progress in Mathematics, Vol. 281 (Birkhäuser, Boston, 2010). CrossrefGoogle Scholar
    • 31. A. Lunardi , Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, Vol. 16 (Birkhäuser, Basel, 1995). CrossrefGoogle Scholar
    • 32. A. Pazy , Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44 (Springer, New York, 1983). CrossrefGoogle Scholar
    • 33. W. Rudin , Real and Complex Analysis, 3rd edn. (McGraw-Hill Book Co., New York, 1987). Google Scholar
    • 34. C. S. Sharma and T. J. Coulson , Spectral theory for unitary operators on a quaternionic Hilbert space, J. Math. Phys. 28 (1987) 1941–1946. Crossref, ISIGoogle Scholar
    • 35. K. Viswanath , Normal operators on quaternionic Hilbert spaces, Trans. Amer. Math. Soc. 162 (1971) 337–350. ISIGoogle Scholar
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