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

V. KhatskevichV. Shul'Man — 1995

Studia Mathematica

The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space ( 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],...

Plus-operators in Krein spaces and dichotomous behavior of irreversible dynamical systems with discrete time

V. KhatskevichL. Zelenko — 2006

Studia Mathematica

We study dichotomous behavior of solutions to a non-autonomous linear difference equation in a Hilbert space. The evolution operator of this equation is not continuously invertible and the corresponding unstable subspace is of infinite dimension in general. We formulate a condition ensuring the dichotomy in terms of a sequence of indefinite metrics in the Hilbert space. We also construct an example of a difference equation in which dichotomous behavior of solutions is not compatible with the signature...

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