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Key wordsIn this paper we study linear fractional relations defined in the following way.Let Hi and H i , i = 1, 2, be Hilbert spaces. We denote the space of bounded linear operators acting from. To each such operator there corresponds a 2 × 2 operator matrix of the formwhere The map GT is called a linear fractional relation.The main result of the paper is the description of operator matrices of the form ( * ) for which the relation GT is defined on some open ball of the space L (H1, H2).Linear fractional relations are natural generalizations of linear fractional transformations studied by M. G. Krein and Yu. L.Šmuljan (1967).The study of both linear fractional transformations and linear fractional relations is motivated by the theory of spaces with an indefinite metric and its applications.

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Inspired by some problems on fractional linear transformations the authors introduce and study the class of operators satisfying the condition ||A|| = max{ρ(AB) : B = 1}, where ρ stands for the spectral radius; and the class of Banach spaces in which all operators satisfy this condition, the authors call such spaces V -spaces. It is shown that many well-known reflexive spaces, in particular, such spaces as L p (0, 1) and C p , are non-V -spaces if p = 2; and that the spaces l p are V -spaces if and only if 1 < p < ∞. The authors pose and discuss some related open problems. Mathematics Subject Classification (2000). Primary 47A10, 47A30; Secondary 47B10, 46B04.

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