Abstract
An operator integral, referred to as the amplitude integral (AI) and used in the BC-method (based on boundarycontrol theory) for solving inverse problems, is systematically studied. For a continuous operator and two families of increasing subspaces, the continual analog of the matrix diagonal in the form of an AI is introduced. The convergence of the AI is discussed. An example of an operator with no diagonal is provided. The role of the diagonal in the problem of triangular factorization is elucidated. The well-known result of matrix theory stating the uniqueness of triangular factorization with a prescribed diagonal is extended. It is shown that the corresponding factor can be represented in the AI form. The correspondence between the AI and the classical representation of the triangular factor of an operator that is a sum of the identity and a compact operator is established.
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References
M. I. Belishev, “Boundary control and continuation of wave fields,” Preprint LOMI, P-1-90, Leningrad (1990).
M. I. Belishev and A. P. Kachalov, “Operator integral in the multidimensional inverse spectral problem,”Zap. Nauchn. Semin. POMI,215, 9–37 (1994).
M. I. Belishev and S. A. Ivanov, “Boundary control and canonical realizations of a two-velocity dynamic system,”Zap. Nauchn. Semin. POMI,222, 18–44 (1995).
I. Ts. Gokhberg and M. G. Krein,The Theory of Volterra Operators in Hilbert Spaces and Its Applications [in Russian], Nauka, Moscow (1967).
M. S. Brodskii,Triangular and Jordan Representations of Linear Operators [in Russian], Nauka, Moscow (1969).
L. A. Sakhnovich, “Problems of factorization and operator identities,”Usp. Mat. Nauk,41, No. 1, 3–55 (1986).
M. I. Belishev, “The canonical model of a dynamic system with boundary control in the inverse problem of heat conductivity,”Algebra Analiz,7, No. 6, 2–32 (1995).
S. A. Avdonin, M. I. Belishev, and S. A. Ivanov, “Controllability in a filled domain for the multidimensional wave equation with a singular boundary control,”Zap. Nauchn. Semin. POMI,210, 7–21 (1994).
M. I. Belishev, V. A. Ryzhov, and V. B. Filippov, “Spectral version of the BC-method; theory and a numerical experiment,”Dokl. Ross. Akad. Nauk,332, No. 4, 414–417 (1994).
L. P. Nizhnik,Inverse Scattering Problems for Hyperbolic Equations [in Russian], Kiev (1991).
Additional information
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 45–60.
This work was supported by INTAS under grant INTAS 93-1815 and by the Russian Foundation for Basic Research under grant 96-01-00666. The authors are grateful to Prof. S. N. Naboko for fruitful discussions and consultations.
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Belishev, M.I., Pushnitskii, A.B. On the triangular factorization of positive operators. J Math Sci 96, 3312–3320 (1999). https://doi.org/10.1007/BF02172806
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DOI: https://doi.org/10.1007/BF02172806