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.
@article{bwmeta1.element.bwnjournal-article-smv137i2p169bwm,
author = {V. Khatskevich},
title = {Generalized fractional linear transformations: convexity and compactness of the image and the pre-image; applications.},
journal = {Studia Mathematica},
volume = {133},
year = {1999},
pages = {169-175},
zbl = {0952.47034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv137i2p169bwm}
}
Khatskevich, V. Generalized fractional linear transformations: convexity and compactness of the image and the pre-image; applications.. Studia Mathematica, Tome 133 (1999) pp. 169-175. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv137i2p169bwm/
[00000] [1] T. Ya. Azizov and I. S. Ǐokhvidov, Foundations of the Theory of Linear Operators in Spaces with Indefinite Metric, Nauka, Moscow, 1986 (in Russian).
[00001] [2] L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer, 1959. | Zbl 0082.07602
[00002] [3] V. Khatskevich, On fixed points of generalized fractional linear transformations, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 1130-1141 (in Russian).
[00003] [4] V. Khatskevich, On the symmetry of properties of a plus-operator and its adjoint operator, Funct. Analysis (Ulyanovsk) 14 (1980), 177-186 (in Russian). | Zbl 0484.47015
[00004] [5] V. Khatskevich, Some global properties of fractional linear transformations, in: Oper. Theory Adv. Appl. 73, Birkhäuser, Basel, 1994, 355-361. | Zbl 0832.47029
[00005] [6] V. Khatskevich and V. Shul'man, Operator fractional linear transformations: convexity and compactness of image; applications, Studia Math. 116 (1995), 189-195.
[00006] [7] 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. | Zbl 0855.34068
[00007] [8] 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. | Zbl 0868.34046
[00008] [9] 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).
[00009] [10] M. A. Krasnosel'skiĭ and A. V. Sobolev, On cones of finite rank, Dokl. Akad. Nauk SSSR 225 (1975), 1256-1259 (in Russian).
[00010] [11] M. G. Kreĭn and Yu. L. Shmul'yan, On fractional-linear transformations with operator coefficients, Mat. Issled. Kishinev 2 (1967), 64-96 (in Russian).