Abstract
We derive various properties, e.g., analyticity of the associated semigroup and existence of a Riesz basis consisting of eigenfunctions, of the operator matrix \( \mathcal{A} = \left[ {\begin{array}{*{20}c} 0 & I \\ { - A_0 } & { - D} \\ \end{array} } \right] \). Here the entries A 0 and D are unbounded operators. Such operator matrices are associated with second-order problems of the form \( \ddot z\left( t \right) + A_0 z\left( t \right) + D\dot z\left( t \right) = 0 \) which are used as models for small motions of some hydrodynamical systems.
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References
T.Ya. Azizov and I.S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, 1989.
T.Ya. Azizov, P. Jonas, and C. Trunk, Spectral points of type π + and π − of self-adjoint operators in Krein spaces, J. Funct. Anal. 226 (2005), 114–137.
H.T. Banks and K. Ito, A unified framework for approximation in inverse problems for distributed parameter systems, Control Theory Adv. Tech. 4 (1988), 73–90.
H.T. Banks, K. Ito, and Y. Wang, Well-posedness for damped second-order systems with unbounded input operators, Differential Integral Equations 8 (1995), 587–606.
J. Behrndt, F. Philipp, and C. Trunk, Properties of the spectrum of type π + and type π − of self-adjoint operators in Krein spaces, Methods Funct. Anal. Topology 12 (2006), 326–340.
C.D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optimization 16 (1978), 373–379.
J. Bognar, Indefinite Inner Product Spaces, Springer, 1974.
S. Chen, K. Liu, and Z. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math. 59 (1999), 651–668.
S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), 15–55.
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer Verlag, New York, 2000.
E. Hendrickson and I. Lasiecka, Numerical approximations and regularizations of Riccati equations arising in hyperbolic dynamics with unbounded control operators, Comput. Optim. Appl. 2 (1993), 343–390.
E. Hendrickson and I. Lasiecka, Finite-dimensional approximations of boundary control problems arising in partially observed hyperbolic systems, Dynam. Contin. Discrete Impuls. Systems 1 (1995), 101–142.
R.O. Hryniv and A.A. Shakalikov, Operator models in the theory of elasticity and in hydrodynamics, and associated analytic semigroups, Moscow Univ. Math. Bull. 54 (1999), 1–10.
F. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim. 26 (1988) 714–724.
I.S. Iohvidov, M.G. Krein, and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Akademie-Verlag, Berlin, 1982.
B. Jacob, K. Morris, and C. Trunk, Minimum-phase infinite-dimensional second-order systems, IEEE Transactions on Automatic Control, 52 (2007), 1654–1665.
B. Jacob, C. Trunk, and M. Winklmeier, Analyticity and Riesz basis property of semigroups associated to damped vibrations, Journal of Evolution Equations, 8 (2008), 263–281.
P. Jonas, On locally definite operators in Krein spaces, in: Spectral Theory and Applications, Theta Ser. Adv. Math. 2 (2003), Theta, Bucharest, 95–127.
T. Kato, Perturbation Theory for Linear Operators, Second Edition, Springer, 1976.
N.D. Kopachevsky, R. Mennicken, Ju.S. Pashkova, and C. Tretter, Complete second-order linear differential operator equations in Hilbert space and applications in hydrodynamics, Trans. Am. Math. Soc. 356 (2004), 4737–4766.
P. Lancaster, A.S. Markus, and V.I. Matsaev, Definitizable operators and quasihyperbolic operator polynomials, J. Funct. Anal. 131 (1995), 1–28.
P. Lancaster, A. Shkalikov, Damped vibrations of beams and related spectral problems, Canadian Applied Mathematics Quarterly 2 (1994), 45–90.
H. Langer, Spektraltheorie linearer Operatoren in J-Räumen und einige Anwendungen auf die Schar \( L\left( \lambda \right) = \lambda ^2 + \lambda B + C \), Habilitationsschrift, Technische Universität Dresden, 1965.
H. Langer, Spectral functions of definitizable operators in Krein spaces, Lect. Notes Math. 948 (1982), 1–46.
H. Langer, A.S. Markus, and V.I. Matsaev, Locally definite operators in indefinite inner product spaces, Math. Ann. 308 (1997), 405–424.
I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations 79 (1989), 340–381.
I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with L 2(0, ∞; L 2(Γ))-feedback control in the Dirichlet boundary condition, J. Differential Equations 66 (1987), 340–390.
N. Levan, The stabilizability problem: A Hilbert space operator decomposition approach, IEEE Trans. Circuits and Systems 25 (1978), 721–727.
N.K. Nikolskii, Treatise on the Shift Operator, Springer 1986.
M. Slemrod, Stabilization of boundary control systems, J. Differential Equation 25 (1976), 402–415.
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Trunk, C. (2009). Analyticity of Semigroups Related to a Class of Block Operator Matrices. In: Grobler, J.J., Labuschagne, L.E., Möller, M. (eds) Operator Algebras, Operator Theory and Applications. Operator Theory: Advances and Applications, vol 195. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0174-0_13
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DOI: https://doi.org/10.1007/978-3-0346-0174-0_13
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