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

The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space ${\displaystyle ℒ(X_1,X_2)}$ of all linear bounded operators acting from ${\displaystyle X_1}$ into ${\displaystyle X_2}$, where ${\displaystyle X_1,X_2}$ are Banach spaces. We show that in the case of Hilbert spaces ${\displaystyle X_1,X_2}$ 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. ${\displaystyle F:K\to {(I+K)}^{-1}}$ the image ${\displaystyle F\left(\right)}$ 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.