Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-17T15:21:57.019Z Has data issue: false hasContentIssue false
  • English
  • Français

On Operator Sum and Product Adjoints and Closures

Published online by Cambridge University Press:  20 November 2018

Karl Gustafson
Karl Gustafson*
Affiliation:
Department of Mathematics, University of Colorado at Boulder, Boulder, CO, U.S.A. e-mail: gustafs@euclid.colorado.edu
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 comment on domain conditions that regulate when the adjoint of the sum or product of two unbounded operators is the sum or product of their adjoints, and related closure issues. The quantum mechanical problem $\text{PHP}$ essentially selfadjoint for unbounded Hamiltonians is addressed, with new results.

Keywords

47A05unbounded operatorsadjoints of sums and productsquantum mechanics
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 3 , 01 September 2011 , pp. 456 - 463
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Albeverio, S., Hoegh-Krohn, R., and Streit, L., Regularization of Hamiltonians and processes. J. Math. Phys. 21(1980), no. 7, 1636–1642.Google Scholar
[2] Gustafson, K., On projections of self-adjoint operators and operator product adjoints. Bull. Amer. Math. Soc. 75(1969), 739–741. doi:10.1090/S0002-9904-1969-12269-XGoogle Scholar
[3] Gustafson, K., Notes on regular positive perturbations. (1972, unpublished).Google Scholar
[4] Gustafson, K., The RKNG (Rellich, Kato, Sz.-Nagy, Gustafson) perturbation theorem for linear operators in Hilbert and Banach Spaces. Acta Sci. Mathe. 45(1983), no. 1–4, 201–211.Google Scholar
[5] Gustafson, K., Operator spectral states. In: Computational tools of complex systems, II, Comput. Math. Appl. 34(1997), no. 5–6, 467–508.Google Scholar
[6] 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
[7] Gustafson, K., Reversibility and regularity. Internat. J. Theoret. Phys. 46(2007), no. 8, 1867–1880. doi:10.1007/s10773-006-9323-9Google Scholar
[8] Gustafson, K., Noncommutative trigonometry and quantum mechanics. In: Advances in deterministic and stochastic analysis, World Sci. Publ., Hackensack, NJ, 2007, pp. 341–360.Google Scholar
[9] Mortad, M. H., On the adjoint and the closure of the sum of two unbounded operators. Canad. Math. Bull. 54(2011) no. 3, 498–505. doi:10.4153/CMB-2011-041-7Google Scholar