Linear fractional relations for Hilbert space operators

In this paper we study linear fractional relations defined in the following way. Let H-i and H'(i), i = 1, 2, be Hilbert spaces. We denote the space of bounded linear operators acting from H-j to H'(i) by L(H-j, H'(i)). Let T is an element of L (H-1 circle plus H-2, H'(1) circle plus H'(2)). To each such operator there corresponds a 2 x 2 operator matrix of the form [GRAPHICS] where T-ij is an element of L(H-j, H-i), i, j = 1, 2. For each such T we define a set-valued map G(T) from L(H-1, H-2) into the set of closed affine subspaces of L(H'(1), H'(2)) by G(T) (K) = {K' is an element of L(H'(1),H'(2)) : T-21 + T-22 K = K'(T-11 + T12K)}. The map GT is called a linearfractional relation. The main result of the paper is the description of operator matrices of the form (*) for which the relation G(T) is defined on some open ball of the space L(H-1, H-2). Linear fractional relations are natural generalizations of linear fractional transformations studied by M. G. Krein and Yu. L. Smuljan (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. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.