Abstract
We consider a J-self-adjoint 2 × 2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry of L is embedded into the absolutely continuous spectrum of the other main-diagonal entry. We work with the analytic continuation of the Schur complement of amain-diagonal entry in L−z to the unphysical sheets of the spectral parameter z plane. We present conditions under which the continued Schur complement has operator roots in the sense of Markus–Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We, then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J-orthogonal invariant subspaces of L. The presentation ends with an explicitly solvable example.
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References
V. M. Adamjan and H. Langer, “Spectral properties of a class of operator-valued functions,” J. Oper. Theory 33, 259–277 (1995).
S. Albeverio, K. A. Makarov, and A. K. Motovilov, “Graph subspaces and the spectral shift function,” Canad. J. Math. 55, 449–503 (2003).
S. Albeverio and A. K. Motovilov, “Operator Stieltjes integrals with respect to a spectral measure and solutions of some operator equations,” Trans. Moscow Math. Soc. 72, 45–77 (2011).
S. Albeverio, A. K. Motovilov, and A. A. Shkalikov, “Bounds on variation of spectral subspaces under J-self-adjoint perturbations,” Integr. Eq. Oper. Theory 64, 455–486 (2009).
S. Albeverio, A. K. Motovilov, and C. Tretter, “Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator,” Indiana Univ. Math. J. 59, 1737–1776 (2010).
T. Y. Azizov, J. Behrndt, P. Jonas, and C. Trunk, “Spectral points of definite type and type p for linear operators and relations in Krein spaces,” J. Lond. Math. Soc. 83), 768–788 (2011).
T. Y. Azizov and I. S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric (John Wiley & Sons, Chichester, 1989).
M. S. Birman and M. Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space (Reidel, Dordrecht, 1987).
J. Bognár, Indefinite Inner Product Spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1974).
K. O. Friedrichs, “On the perturbation of continuous spectra,” Comm. Pure Appl. Math. 1, 361–406 (1948).
V. Hardt, R. Mennicken, and A. K. Motovilov, “Factorization theorem for the transfer function associated with a 2 × 2 operator matrix having unbounded couplings,” J. Oper. Theory 48, 187–226 (2002).
V. Hardt, R. Mennicken, and A. K. Motovilov, “Factorization theorem for the transfer function associated with an unbounded non–self-adjoint 2×2 operator matrix,” Oper. Theory: Adv. Appl. 142, 117–132 (2003).
T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, New York, 1966).
H. Langer, “Spectral functions of definitizable operators in Krein spaces,” Lect. Notes Math. 948, 1–46 (1982).
A. S. Markus and V. I. Matsaev, “On the spectral theory of holomorphic operator-functions in Hilbert space,” Funct. Anal. Appl. 9, 73–74 (1975).
R. Mennicken and A. K. Motovilov, “Operator interpretation of resonances arising in spectral problems for 2 × 2 operator matrices,” Math. Nachr. 201, 117–181 (1999).
R. Mennicken and A. K. Motovilov, “Operator interpretation of resonances arising in spectral problems for 2 × 2 matrix Hamiltonians,” Oper. Theory: Adv. Appl. 108, 316–322 (1999).
R. Mennicken and A. A. Shkalikov, “Spectral decomposition of symmetric operator matrices,” Math. Nachr. 179, 259–273 (1996).
A. K. Motovilov, “Removal of the resolvent-like energy dependence from interactions and invariant subspaces of a total Hamiltonian,” J. Math. Phys. 36, 6647–6664 (1995).
F. Philipp, V. Strauss, and C. Trunk, “Local spectral theory for normal operators in Krein spaces,” Math. Nach. 286, 42–58 (2013).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV: Analysis of Operators (Academic Press, New York, 1978).
C. Tretter, Spectral Theory of Block Operator Matrices and Applications (Imperial College Press, London, 2008).
C. Tretter, “Spectral inclusion for unbounded block operator matrices,” J. Funct. Anal. 256, 3806–3829 (2009).
K. Veselić, “On spectral properties of a class of J-selfadjoint operators. I,” GlasnikMat. 7, 229–248 (1972).
K. Veselić, “On spectral properties of a class of J-selfadjoint operators. II,” GlasnikMat. 7, 249–254 (1972).
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Albeverio, S., Motovilov, A.K. On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case. Math Notes 100, 761–773 (2016). https://doi.org/10.1134/S0001434616110158
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DOI: https://doi.org/10.1134/S0001434616110158