Abstract
Let\(\mathfrak{G}\) be a connected, finite-dimensional, complex analytic manifold; let T(λ) be an analytic function defined on\(\mathfrak{G}\), whose values are J-biexpanding operators on a J-space H. Let\(\Re \)(A) denote the range of A. The following assertions are proved: 1. The lineals\(\Re (\sqrt {T(\lambda )*J T (\lambda ) - J} ) \equiv \Re \) and\(\Re (\sqrt {T(\lambda )J T (\lambda )* - J} ) \equiv \Re \) do not depend on λ. 2. For arbitrary λ μ ∈\(\mathfrak{G}\) we have\(\Re ({\rm T} (\lambda ) - T (\mu )) \subset \Re _ * , \Re ({\rm T} (\lambda ) * - {\rm T} (\mu )^* ) \subset \Re \)
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Literature cited
Yu. P. Ginzburg, “On projections in a Hilbert space with a bilinear metric,” Dokl. Akad. Nauk SSSR,139, No. 4, 775–778 (1961).
R. G. Doublas, “On majorization, factorization, and range inclusion of operators on a Hilbert space,” Proc. Amer. Math. Soc.,17, No. 2, 413–415 (1966).
Yu. L. Shmul'yan, “Two-sided division in a ring of operators,” Matem. Zametki,1, No.5, 605–610 (1967).
Yu. P. Ginzburg, “The maximum principle for J-nonexpanding operator functions and some corollaries,” Izv. Vyssh. Uchebn. Zaved.,1, 42–53 (1963).
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York (1966).
Yu. P. Ginzburg and I. S. Iokhvidov, “Investigation of the geometry of infinite-dimensional spaces with a bilinear metric,” Usp. Matem. Nauk,17, No. 4, 3–56 (1962).
M. G. Krein, Introduction to the Geometry of Indefinite J-spaces and the Theory of Operators in These Spaces [in Russian], Vtoraya Letn. Matem. Shkola, I, Naukova Dumka, Kiev (1965), pp. 15–92.
Yu. P. Ginzburg, “J-nonexpanding operators on a Hilbert space,” Nauchn. Zapiski Odessk. Ped. In-ta,22, No. 1, 13–19 (1958).
A. V. Shtraus, “On a class of regular operator functions,” Dokl. Akad. Nauk SSSR,70, No. 4, 577–580 (1950).
B. S. Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland (1971).
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Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 511–520, October, 1976.
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Shmul'yan, Y.L. Some stability properties for analytic operator functions. Mathematical Notes of the Academy of Sciences of the USSR 20, 843–848 (1976). https://doi.org/10.1007/BF01098900
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DOI: https://doi.org/10.1007/BF01098900