Abstract
We consider some properties of fractional-linear transformations (such as focusedness and compactness of the image) which may be applied to the study behavior of solution of evolution problems (see, for example, [1], [2], [3]). Consider a solution x(t) = U(t)x 0, U(0) = I, of an evolution problem in a Hilbert space ℌ with the scalar product (·,·). The space ℌ is endowed with the corresponding indefinite metric V(t), the operator U(t) is a plus-operator (see below) with respect to V(t): (V(t)U(t)x, U(t)x) ≥ 0 for all x ∊ ℌ such that (V(t)x,x) ≥ 0. The study of the behavior of x(t) is naturally divided into the following three cases: 1) U(t) is invertible, i.e. (U(t))−1 exists, is defined on the whole of ℌ and is bounded (hyperbolic); 2) U(t) is bounded (parabolic); 3) U(t) is unbounded (elliptic).
Supported in part by the Ministry of Absorption and the Rashi Foundation
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Dedicated to Professor M. Livsic
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Khatskevich, V. (1994). Some Global Properties of Fractional-Linear Transformations. In: Feintuch, A., Gohberg, I. (eds) Nonselfadjoint Operators and Related Topics. Operator Theory: Advances and Applications, vol 73. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8522-5_13
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DOI: https://doi.org/10.1007/978-3-0348-8522-5_13
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