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

V. Khatskevich; V. Shul'Man

Studia Mathematica (1995)

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

Abstract

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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.

How to cite

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Khatskevich, 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

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  1. [1] L. Cesary, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer, 1959. 
  2. [2] R. C. James, Orthogonality in normed linear spaces, Duke Math. J. 12 (1945), 291-302. Zbl0060.26202
  3. [3] V. Khatskevich, Some global properties of operator fractional-linear transformations, in: Proc. Israel Mathematical Union Conference, Beer-Sheva 1993, 17-18. 
  4. [4] V. Khatskevich, Global properties of fractional-linear transformations, in: Operator Theory, Birkhäuser, Basel, 1994, 355-361. Zbl0832.47029
  5. [5] V. Khatskevich and L. Zelenko, Indefinite metrics and dichotomy of solutions for linear differential equations in Hilbert spaces, preprint, 1993. Zbl0855.34068
  6. [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. [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. [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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