Adjoints and formal adjoints of matrices of unbounded operators

Möller, Manfred; Szafraniec, Franciszek

doi:10.1090/S0002-9939-08-09211-3

Adjoints and formal adjoints of matrices of unbounded operators
HTML articles powered by AMS MathViewer

by Manfred Möller and Franciszek Hugon Szafraniec PDF

Proc. Amer. Math. Soc. 136 (2008), 2165-2176 Request permission

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.

References

F. V. Atkinson, H. Langer, R. Mennicken, and A. A. Shkalikov, The essential spectrum of some matrix operators, Math. Nachr. 167 (1994), 5–20. MR 1285306, DOI 10.1002/mana.19941670102
J. A. Burns and T. L. Herdman, Adjoint semigroup theory for a class of functional differential equations, SIAM J. Math. Anal. 7 (1976), no. 5, 729–745. MR 417528, DOI 10.1137/0507056
Wolfgang Desch, Ronald Grimmer, and Wilhelm Schappacher, Well-posedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations 74 (1988), no. 2, 391–411. MR 952904, DOI 10.1016/0022-0396(88)90011-3
Klaus-J. Engel, Matrix representation of linear operators on product spaces, Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 219–224. International Workshop on Operator Theory (Cefalù, 1997). MR 1710840
J. P. Friedberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987.
Volker Hardt, Alexei Konstantinov, and Reinhard Mennicken, On the spectrum of the product of closed operators, Math. Nachr. 215 (2000), 91–102. MR 1768195, DOI 10.1002/1522-2616(200007)215:1<91::AID-MANA91>3.3.CO;2-Y
Volker Hardt, Reinhard Mennicken, and Alexander 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
K. Hain and R. Lüst, Zur Stabilität zylindersymmetrischer Plasmakonfigurationen mit Volumenströmen, Z. Naturf. 13a (1958), 936–940 (German). MR 0106671, DOI 10.1515/zna-1958-1103
Seppo Hassi, Adrian Sandovici, Henk de Snoo, and Henrik Winkler, Extremal extensions for the sum of nonnegative selfadjoint relations, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3193–3204. MR 2322750, DOI 10.1090/S0002-9939-07-08827-2
Alexei Konstantinov and Reinhard Mennicken, On the Friedrichs extension of some block operator matrices, Integral Equations Operator Theory 42 (2002), no. 4, 472–481. MR 1885445, DOI 10.1007/BF01270924
Pavel Kurasov and Serguei Naboko, On the essential spectrum of a class of singular matrix differential operators. I. Quasiregularity conditions and essential self-adjointness, Math. Phys. Anal. Geom. 5 (2002), no. 3, 243–286. MR 1940117, DOI 10.1023/A:1020929007538
Alexander E. Lifschitz, Magnetohydrodynamics and spectral theory, Developments in Electromagnetic Theory and Applications, vol. 4, Kluwer Academic Publishers Group, Dordrecht, 1989. MR 990647, DOI 10.1007/978-94-009-2561-8
Rainer Nagel, Towards a “matrix theory” for unbounded operator matrices, Math. Z. 201 (1989), no. 1, 57–68. MR 990188, DOI 10.1007/BF01161994
Schôichi Ôta and K. Schmüdgen, Some selfadjoint $2\times 2$ operator matrices associated with closed operators, Integral Equations Operator Theory 45 (2003), no. 4, 475–484. MR 1971749, DOI 10.1007/s000200300017
Franciszek Hugon Szafraniec, On normal extensions of unbounded operators. IV. A matrix construction, Operator theory and indefinite inner product spaces, Oper. Theory Adv. Appl., vol. 163, Birkhäuser, Basel, 2006, pp. 337–350. MR 2215869, DOI 10.1007/3-7643-7516-7_{1}4