Operator fractional-linear transformations: convexity and compactness of image; applications
Khatskevich, V. ; Shul'Man, V.
Studia Mathematica, Tome 113 (1995), p. 189-195 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space ℒ(X1,X2) of all linear bounded operators acting from X1 into X2, where X1,X2 are Banach spaces. We show that in the case of Hilbert spaces X1,X2 the image F(ℬ) of any (open or closed) ball ℬ ⊂ D(F) is convex, and if ℬ is closed, then F(ℬ) is compact in the weak operator topology (w.o.t.) (Theorem 1.2). These results extend the corresponding results on compactness obtained in [3], [4] under some additional restrictions imposed on F. We also establish that the convexity of the image of f.-l.t. is a characteristic property of Hilbert spaces, that is, if for the f.-l.t. F:K→(I+K)-1 the image F() of the open unit ball of the space ℒ(X) is convex, then X is a Hilbert space (Theorem 1.3). In Section 2 we apply the compactness of F(̅) for the closed unit operator ball ̅ to the study of the behavior of solutions to evolution problems in a Hilbert space ℋ. Namely, we establish the exponential dichotomy of solutions for the so-called hyperbolic case (such that the evolution operator is invertible). This is an extension of Theorem 1.1 of [5], where the corresponding assertion was established for the particular case of a Pontryagin space ℋ.

Publié le : 1995-01-01
EUDML-ID : urn:eudml:doc:216226 
@article{bwmeta1.element.bwnjournal-article-smv116i2p189bwm,
     author = {V. Khatskevich and V. Shul'Man},
     title = {Operator fractional-linear transformations: convexity and compactness of image; applications},
     journal = {Studia Mathematica},
     volume = {113},
     year = {1995},
     pages = {189-195},
     zbl = {0842.47022},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv116i2p189bwm}
}

Khatskevich, V.; Shul'Man, V. Operator fractional-linear transformations: convexity and compactness of image; applications. Studia Mathematica, Tome 113 (1995) pp. 189-195. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv116i2p189bwm/

[00000] [1] L. Cesary, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer, 1959.

[00001] [2] R. C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291-302. | Zbl 0060.26202

[00002] [3] V. Khatskevich, Some global properties of operator fractional-linear transformations, in: Proc. Israel Mathematical Union Conference, Beer-Sheva 1993, 17-18.

[00003] [4] V. Khatskevich, Global properties of fractional-linear transformations, in: Operator Theory, Birkhäuser, Basel, 1994, 355-361. | Zbl 0832.47029

[00004] [5] V. Khatskevich and L. Zelenko, Indefinite metrics and dichotomy of solutions for linear differential equations in Hilbert spaces, preprint, 1993. | Zbl 0855.34068

[00005] [6] M. A. Krasnosel'skiĭ and A. V. Sobolev, On cones of finite rank, Dokl. Akad. Nauk SSSR 225 (1975), 1256-1259 (in Russian).

[00006] [7] M. G. Kreĭn and Yu. L. Shmul'yan, On fractional-linear transformations with operator coefficients, Mat. Issled. (Kishinev) 2 (1967), 64-96 (in Russian).

[00007] [8] Yu. L. Shmul'yan, On divisibility in the class of plus-operators, Mat. Zametki 74 (1967), 516-525 (in Russian).