Hostname: page-component-5b777bbd6c-gcwzt Total loading time: 0 Render date: 2025-06-24T01:30:36.067Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Adjoint and the Closure of the Sum of Two Unbounded Operators

Published online by Cambridge University Press:  20 November 2018

Mohammed Hichem Mortad
Mohammed Hichem Mortad*
Affiliation:
Department of Mathematics, University of Oran Es-Senia, BP 1524 El Menouar, 31000 Oran, Algeria, Mailing Address: BP 7085 Seddikia, Oran 31013, Algeria e-mail: mhmortad@gmail.commortad@univ-oran.dz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove, under some conditions on the domains, that the adjoint of the sum of two unbounded operators is the sum of their adjoints in both Hilbert and Banach space settings. A similar result about the closure of operators is also proved. Some interesting consequences and examples “spice up” the paper.

Keywords

47A05unbounded operatorssum and products of operatorsHilbert and Banach adjointsself-adjoint operatorsclosed operatorsclosure of operators
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 3 , 01 September 2011 , pp. 498 - 505
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Albeverio, S., Høegh-Krohn, R., and Streit, L., Regularization of Hamiltonians and processes. J. Math. Phys. 21(1980), no. 7, 1636–1642. doi:10.1063/1.524649Google Scholar
[2] Beals, R. W., A note on the adjoint of a perturbed operator. Bull. Amer. Math. Soc. 70(1964), 314–315. doi:10.1090/S0002-9904-1964-11137-XGoogle Scholar
[3] van Casteren, J. A. W., Adjoints of products of operators in Banach space. Arch. Math. (Basel) 23(1972), 73–76.Google Scholar
[4] van Casteren, J. A. W. and Goldberg, S., The conjugate of the product of operators. Studia Math. 38(1970), 125–130.Google Scholar
[5] Diagana, T., Schrödinger operators with a singular potential. Int. J. Math. Math. Sci. 29(2002), no. 6, 371–373. doi:10.1155/S0161171202007330Google Scholar
[6] Diagana, T., A generalization related to Schrödinger operators with a singular potential. Int. J. Math. Math. Sci. 29(2002), no. 10, 609–611. doi:10.1155/S0161171202007974Google Scholar
[7] Dixmier, J., L’adjoint du produit de deux Opérateurs Fermés. Ann. Fac. Sci. Univ. Toulouse (4) 11(1947), 101–106.Google Scholar
[8] Dore, G. and Venni, A., On the closedness of the sum of two closed operators. Math. Z. 196(1987), no. 2, 189–201. doi:10.1007/BF01163654Google Scholar
[9] Goldberg, S., Unbounded linear operators: Theory and applications. McGraw-Hill, New York-Toronto-London, 1966.Google Scholar
[10] Gustafson, K., On projections of selfadjoint operators and operator product adjoints. Bull. Amer. Math. Soc. 75(1969), 739–741. doi:10.1090/S0002-9904-1969-12269-XGoogle Scholar
[11] Gustafson, K., A composition adjoint lemma. In: Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999), CMS Conf. Proc., 29, American Mathematical Society, Providence, RI, 2000, pp. 253–258.Google Scholar
[12] Hess, P. and Kato, T., Perturbation of closed operators and their adjoints. Comment. Math. Helv. 45(1970), 524–529. doi:10.1007/BF02567350Google Scholar
[13] Holland, S. S. Jr., On the adjoint of the product of operators. J. Functional Analysis, 3(1969), 337–344. doi:10.1016/0022-1236(69)90029-9Google Scholar
[14] Kato, T., Perturbation theory for linear operators. Grundlehren der Mathematischen Wissenschaften, 132, Springer-Verlag, Berlin-New York, 1976.Google Scholar
[15] Kosaki, H., On intersections of domains of unbounded positive operators. Kyushu J. Math. 60(2006) no. 1, 3–25. doi:10.2206/kyushujm.60.3Google Scholar
[16] Labbas, R., Some results on the sum of linear operators with nondense domains. Ann. Mat. Pura Appl. 154(1989), 91–97. doi:10.1007/BF01790344Google Scholar
[17] Meise, R. and Vogt, D., Introduction to functional analysis. Oxford Graduate Texts in Mathematics, 2, The Clarendon Press, Oxford University Press, New York, 1997.Google Scholar
[18] Messirdi, B. and Mortad, M. H., On different products of closed operators. Banach J. Math. Anal. 2(2008), no. 1, 40–47.Google Scholar
[19] Mortad, M. H., An application of the Putnam-Fuglede theorem to normal products of self-adjoint operators. Proc. Amer. Math. Soc. 131(2003), no. 10, 3135–3141. doi:10.1090/S0002-9939-03-06883-7Google Scholar
[20] Mortad, M. H., Self-adjointness of the perturbed wave operator on L 2(ℝ n ), n ≥ 2 . Proc. Amer. Math. Soc. 133(2005), no. 2, 455–464. doi:10.1090/S0002-9939-04-07552-5Google Scholar
[21] Mortad, M. H., On Lp-estimates for the time-dependent Schrödinger operator on L 2 . J. Inequal Pure Appl. Math. 8(2007), no. 3, Article 80, 8pp.Google Scholar
[22] Mortad, M. H., On the closedness, the self-adjointness and the normality of the product of two closed operators. Demonstratio Math., to appear.Google Scholar
[23] von Neumann, J., Zur Theorie des unbeschränkten Matrizen. J. Reine Angew. Math. 161(1929), 208–236.Google Scholar
[24] Reed, M. and Simon, B., Methods of modern mathematical physics.I. Functional analysis. Academic Press, New York-London, 1972.Google Scholar
[25] Reed, M. and Simon, B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York-London, 1975.Google Scholar
[26] Schechter, M., The conjugate of a product of operators. J. Functional Analysis, 6(1970), 26–28. doi:10.1016/0022-1236(70)90045-5Google Scholar
[27] Schmüdgen, K., On domains of powers of closed symmetric operators. J. Operator Theory 9(1983), no. 1, 53–75.Google Scholar
[28] Sz.-Nagy, B., Perturbations des transformations linéaires fermées. Acta Sci. Math. Szeged 14(1951), 125–137.Google Scholar
[29] Weidmann, J., Linear operators in Hilbert spaces. Graduate Texts in Mathematics, 68, Springer-Verlag, New York-Berlin, 1980.Google Scholar