Abstract
In this paper several classes of self-adjoint and non-self-adjoint block operator matrices with unbounded entries are studied. The main results concern the existence of maximal spectral invariant subspaces which correspond to the right and the left half planes and their representation by means of angular operators. Sometimes this yields a diagonalization of the block operator matrix under consideration. Applications to abstract Dirac operators with supersymmetry and to Dirac operators with a potential are given.
H. Langer was supported by the “Fonds zur Forderung der wissenschaftlichen Forschung” of Austria, Project P 12176-MAT. C. Tretter gratefully acknowledges the support of the German Research Foundation, DFG, Grant TR 368/4–1
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References
V.M. Adamjan and H. Langer, Spectral properties of a class of rational operator valued functions; J. Operator Theory 33(1995), 259–277.
V.M. Adamjan, H. Langer, R. Mennicken and J. Saurer, Spectral components of selfadjoint block operator matrices with unbounded entries; Math. Nachr. 178(1996), 43–80.
T.Ya. Azizov and I.S. Iohvidov, Linear Operators in Spaces with an Indefinite Metric;John Wiley & Sons, Chichester New York Brisbane Toronto Singapore,1989.
H. Bart, I.C. Gohberg and M.A. Kaashoek, Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators; J. Funct. Anal. 68(1986), 1–42.
K.-J. Engel, Operator Matrices and Systems of Evolution Equations; Manuscript, Tuebingen, 1996.
I.C. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators vol. I; Operator Theory: Adv. Appl. 49, Birkhäuser, Basel, 1990.
M. Griesemer and H. Siedentop, A minimax principle for the eigenvalues in gaps; J. London Math. Soc., to appear.
I.S. Iohvidov and M.G. Krein, Spectral theory of operators in spaces with an indefinite metric, I. Amer. Math. Soc. Transl. (2) 13(1960), 105–175.
I.S. Iohvidov, M.G. Krein and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric; Math. Researchvol. 9, AkademieVerlag, Berlin, 1982.
M.A. Kaashoek and S.M. Verduyn-Lunel, An Integrability Condition on the Resolvent for Hyperbolicity of the Semigroup; J. Differential Equations 112:2(1994), 374–406.
T. Kato, Perturbation Theory for Linear Operators; Springer Verlag, Berlin Heidelberg New York, 1966.
H. Langer and C. Tretter, Spectral decomposition of some nonselfadjoint block operator matrices; J. Operator Theory 39(1998), 339–359.
H. Langer, A. Markus, V. Matsaev and C. Tretter, A new concept for block operator matrices: The quadratic numerical range; Math. Ann., submitted.
A.K. Motovilov, Removal of the energy dependence from the resolvent-like energy-dependent interactions, preprint JINR E5–94–259, Dubna, 1994.
R. Mennicken and A.A. Shkalikov, Spectral decomposition of symmetric opera-tor matrices; Math. Nachr. 179(1996), 259–273.
B. Thaller, The Dirac Equation; Springer-Verlag, Berlin Heidelberg New York,1992.
K. Veselič, On spectral properties of a class of J-selfadjoint operators, I; Glas. Mat. Ser. III, 7(1972), 229–248.
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Langer, H., Tretter, C. (2001). Diagonalization of Certain Block Operator Matrices and Applications to Dirac Operators. In: Bart, H., Ran, A.C.M., Gohberg, I. (eds) Operator Theory and Analysis. Operator Theory: Advances and Applications, vol 122. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8283-5_13
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DOI: https://doi.org/10.1007/978-3-0348-8283-5_13
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