Linear fractional relations in Banach spaces: interior points in the domain and analogues of the Liouville theorem

In this paper we study linear fractional relations defined in the following way. Let Bi, B′ i, i = 1, 2, be Banach spaces. We denote the space of bounded linear operators by L. Let T ∈ L(B1 ⊕ B2,B 1 ⊕ B′ 2). To each such operator there corresponds a 2 × 2 operator matrix of the form T = ( T11 T12 T21 T22 ) , (∗) where Tij ∈ L(Bj,B i), i, j = 1, 2. For each such T we define a set-valued map GT from L(B1,B2) into the set of closed affine subspaces of L(B′ 1,B 2) by GT (K) = {K ′ ∈ L(B′ 1,B 2) : T21 + T22K = K (T11 + T12K)}. The map GT is called a linear fractional relation. The paper is devoted to the following two problems: • Characterization of operator matrices of the form (*) for which the set GT (K) is non-empty for each K in some open ball of the space L(B1,B2). • Characterizations of quadruples (B1,B2,B 1,B 2) of Banach spaces such that linear fractional relations defined for such spaces satisfy the natural analogue of the Liouville theorem “a bounded entire function is constant”. 2000 Mathematics Subject Classification: 47A56, 46B20, 47B50. ∗Supported by St. John’s University Summer 2006 Support of Research Program The author thanks V. A. Khatskevich for suggesting him the problems considered in this paper, V. S. Shulman for his advice and encouragement, and the referee for his helpful critical comments.