Geodesic
Operator fractional-linear transformations: convexity and compactness of image; applications
Studia Mathematica, Tome 116 (1995) no. 2, pp. 189-195

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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(
DOI : 10.4064/sm-116-2-189-195
Keywords: Hilbert space, fractional-linear transformation, evolution operator, indefinite metric

V. Khatskevich 1

1 
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V. Khatskevich. Operator fractional-linear transformations: convexity and compactness of image; applications. Studia Mathematica, Tome 116 (1995) no. 2, pp. 189-195. doi: 10.4064/sm-116-2-189-195

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