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Adjoints and formal adjoints of matrices of unbounded operators

Authors: Manfred Möller and Franciszek Hugon Szafraniec
Journal: Proc. Amer. Math. Soc. 136 (2008), 2165-2176
MSC (2000): Primary 47A05; Secondary 47D06
Published electronically: February 14, 2008
MathSciNet review: 2383522
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Abstract: In this paper we discuss diverse aspects of the mutual relationship between adjoints and formal adjoints of unbounded operators bearing a matrix structure. We emphasize the behaviour of row and column operators as they turn out to be the germs of an arbitrary matrix operator, providing most of the information about the latter as it is the troublemaker.

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  • 1. F. V. Atkinson, H. Langer, R. Mennicken, A. A. Shkalikov, The essential spectrum of some matrix operators, Math. Nachr. 167 (1994), 5-22. MR 1285306 (95f:47007)
  • 2. J. A. Burns, T. L. Herdman, Adjoint semigroup theory for a class of functional differential equations, SIAM J. Math. Anal. 7 (1976), 729-745. MR 0417528 (54:5578)
  • 3. W. Desch, R. Grimmer, W. Schappacher, Wellposedness and wave propagation for a class of integrodifferential equations in Banach space, J. Diff. Equ. 74 (1988), 391-411. MR 952904 (89e:45012)
  • 4. K.-L. Engel, Matrix representation of linear operators on product spaces, Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 219-224. MR 1710840 (2000f:47004)
  • 5. J. P. Friedberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987.
  • 6. V. Hardt, A. Konstantinov, R. Mennicken, S. Naboko, On the spectrum of the product of closed operators, Math. Nachr 215 (2000), 91-102. MR 1768195 (2001b:47008)
  • 7. V. Hardt, R. Mennicken, A. K. Motovilov, A factorization theorem for the transfer function associated with a $ 2\times 2$ operator matrix having unbounded couplings, J. Operator Theory 48 (2002), no. 1, 187-226. MR 1926050 (2003g:47026)
  • 8. K. Hain, R. Lüst, Zur Stabilität zylindersymmetrischer Plasmakonfigurationen mit Volumenströmen, Z. Naturforsch. A 13 (1958), 936-940. MR 0106671 (21:5402)
  • 9. S. Hassi, A. Sandovici, H. de Snoo and H. Winkler, Extremal extensions for the sum of nonnegative selfadjoint relations, Proc. Amer. Math. Soc., 135 (2007), 3193-3204. MR 2322750.
  • 10. A. Konstantinov, R. Mennicken, On the Friedrichs extension of some block operator matrices, Integral Equations Operator Theory 42 (2002), 472-481. MR 1885445 (2003m:47039)
  • 11. P. Kurasov, S. Naboko, On the essential spectrum of a class of singular matrix differential operators. I: Quasiregularity conditions and essential self-adjointness, Mathematical Physics, Analysis and Geometry 5 (2002), 243-286. MR 1940117 (2003m:47084)
  • 12. A. Lifschitz, Magnetohydrodynamics and Spectral Theory, Kluwer, Dordrecht, 1989. MR 990647 (90k:76097)
  • 13. R. Nagel, Towards a ``matrix theory'' for unbounded operator matrices, Math. Z. 201 (1989), 57-68. MR 990188 (90c:47004)
  • 14. S. Ôta, K. Schmüdgen, Some selfadjoint $ 2\times 2$ operator matrices associated with closed operators, Integr. Equ. Oper. Theory 45 (2003), 475-484. MR 1971749 (2004b:47041)
  • 15. F.H. Szafraniec, On normal extensions of unbounded operators: IV. A matrix construction, in: Operator Theory: Adv. Appl. 163 (2005), 337-350. MR 2215869 (2007b:47052)

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Additional Information

Manfred Möller
Affiliation: The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, WITS, 2050, South Africa

Franciszek Hugon Szafraniec
Affiliation: Instytut Mathematyki, Uniwersytet Jagielloński, ul. Reymonta 4, Pl-30059 Kraków, Poland

Keywords: Operator matrix, row operator, column operator, adjoint and formal adjoint
Received by editor(s): December 11, 2006
Received by editor(s) in revised form: April 27, 2007
Published electronically: February 14, 2008
Additional Notes: This work was supported by a grant of the NRF of South Africa, GUN 2053746, and by the grant KBN 2 P 03A 637 024 (Poland).
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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