# Operator fractional-linear transformations: convexity and compactness of image; applications

Studia Mathematica (1995)

- Volume: 116, Issue: 2, page 189-195
- ISSN: 0039-3223

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topKhatskevich, V., and Shul'Man, V.. "Operator fractional-linear transformations: convexity and compactness of image; applications." Studia Mathematica 116.2 (1995): 189-195. <http://eudml.org/doc/216226>.

@article{Khatskevich1995,

abstract = {The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space $ℒ(X_1,X_2)$ of all linear bounded operators acting from $X_1$ into $X_2$, where $X_1, X_2$ are Banach spaces. We show that in the case of Hilbert spaces $X_1, X_2$ 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.},

author = {Khatskevich, V., Shul'Man, V.},

journal = {Studia Mathematica},

keywords = {Hilbert space; fractional-linear transformation; evolution operator; indefinite metric; fractional-linear transformations; weak operator topology; convexity of the image; behavior of solutions to evolution problems in a Hilbert space; exponential dichotomy; hyperbolic case; Pontryagin space},

language = {eng},

number = {2},

pages = {189-195},

title = {Operator fractional-linear transformations: convexity and compactness of image; applications},

url = {http://eudml.org/doc/216226},

volume = {116},

year = {1995},

}

TY - JOUR

AU - Khatskevich, V.

AU - Shul'Man, V.

TI - Operator fractional-linear transformations: convexity and compactness of image; applications

JO - Studia Mathematica

PY - 1995

VL - 116

IS - 2

SP - 189

EP - 195

AB - The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space $ℒ(X_1,X_2)$ of all linear bounded operators acting from $X_1$ into $X_2$, where $X_1, X_2$ are Banach spaces. We show that in the case of Hilbert spaces $X_1, X_2$ 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.

LA - eng

KW - Hilbert space; fractional-linear transformation; evolution operator; indefinite metric; fractional-linear transformations; weak operator topology; convexity of the image; behavior of solutions to evolution problems in a Hilbert space; exponential dichotomy; hyperbolic case; Pontryagin space

UR - http://eudml.org/doc/216226

ER -

## References

top- [1] L. Cesary, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer, 1959.
- [2] R. C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291-302. Zbl0060.26202
- [3] V. Khatskevich, Some global properties of operator fractional-linear transformations, in: Proc. Israel Mathematical Union Conference, Beer-Sheva 1993, 17-18.
- [4] V. Khatskevich, Global properties of fractional-linear transformations, in: Operator Theory, Birkhäuser, Basel, 1994, 355-361. Zbl0832.47029
- [5] V. Khatskevich and L. Zelenko, Indefinite metrics and dichotomy of solutions for linear differential equations in Hilbert spaces, preprint, 1993. Zbl0855.34068
- [6] M. A. Krasnosel'skiĭ and A. V. Sobolev, On cones of finite rank, Dokl. Akad. Nauk SSSR 225 (1975), 1256-1259 (in Russian).
- [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).
- [8] Yu. L. Shmul'yan, On divisibility in the class of plus-operators, Mat. Zametki 74 (1967), 516-525 (in Russian).

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