ArticleOriginal scientific text
Title
Generalized fractional linear transformations: convexity and compactness of the image and the pre-image; applications.
Authors 1
Affiliations
- Department of Applied Mathematics, Ort Braude College, P.O. Box 78, 21982 Karmiel, Israel
Abstract
The convexity and compactness in the weak operator topology of the image and pre-image of a generalized fractional linear transformation is established. As an application the exponential dichotomy of solutions to evolution problems of the parabolic type is proved.
Bibliography
- T. Ya. Azizov and I. S. Ǐokhvidov, Foundations of the Theory of Linear Operators in Spaces with Indefinite Metric, Nauka, Moscow, 1986 (in Russian).
- L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer, 1959.
- V. Khatskevich, On fixed points of generalized fractional linear transformations, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 1130-1141 (in Russian).
- V. Khatskevich, On the symmetry of properties of a plus-operator and its adjoint operator, Funct. Analysis (Ulyanovsk) 14 (1980), 177-186 (in Russian).
- V. Khatskevich, Some global properties of fractional linear transformations, in: Oper. Theory Adv. Appl. 73, Birkhäuser, Basel, 1994, 355-361.
- V. Khatskevich and V. Shul'man, Operator fractional linear transformations: convexity and compactness of image; applications, Studia Math. 116 (1995), 189-195.
- V. Khatskevich and L. Zelenko, Indefinite metrics and dichotomy of solutions to linear differential equations in Hilbert spaces, Chinese J. Math. 2 (1996), 99-112.
- V. Khatskevich and L. Zelenko, The fractional-linear transformations of the operator ball and dichotomy of solutions to evolution equations, in: Contemp. Math. 204, Amer. Math. Soc., 1997, 149-154.
- V. A. Khatskevich and A. V. Sobolev, On definite invariant subspaces and spectral structure of focused plus-operators, Funktsional. Anal. i Prilozhen. 15 (1981), no. 1, 84-85 (in Russian).
- M. A. Krasnosel'skiĭ and A. V. Sobolev, On cones of finite rank, Dokl. Akad. Nauk SSSR 225 (1975), 1256-1259 (in Russian).
- M. G. Kreĭn and Yu. L. Shmul'yan, On fractional-linear transformations with operator coefficients, Mat. Issled. Kishinev 2 (1967), 64-96 (in Russian).