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

• Autorzy: V. Khatskevich , V. Shul'Man
• Języki publikacji: EN

Abstrakty

EN

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 $F(𝘒)$ of the open unit ball 𝘒 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(𝘒̅) for the closed unit operator ball 𝘒̅ 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.

Słowa kluczowe

EN

Hilbert space; fractional-linear transformation; evolution operator; indefinite metric

Kategorie tematyczne

• 47A53: (Semi-) Fredholm operators; index theories
• 47B50: Operators on spaces with an indefinite metric

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1995
• Tom: 116
• Numer: 2
• Strony: 189-195

Daty

wydano

1995

otrzymano

1995-01-19

poprawiono

1995-05-16

Twórcy

autor

V. Khatskevich

• Department of Applied Mathematics, International College of Technology, P.O. Box 78, Karmiel 20101, Israel

autor

V. Shul'Man

• Department of Mathematics, Vologda Polytechnical Institute, 15 Lenin st., 160008 Vologda, Russia

Bibliografia

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