Abstract
In this paper we study linear fractional relations defined in the following way.
Let βοΈ~i~ and βοΈ~i~ ^β²^, i = 1, 2, be Hilbert spaces. We denote the space of bounded linear operators acting from βοΈ~j~ to βοΈ~i~ ^β²^ by L (βοΈ~j~ , βοΈ~i~ ^β²^). Let T β βοΈ(βοΈ~1~ β βοΈ~2~, βοΈ~1~^β²^ β βοΈ~2~^β²^). To each such operator there corresponds a 2 Γ 2 operator matrix of the form
equation image
where T~ij~ β βοΈ (βοΈ~j~ , βοΈ~i~ ^β²^), i, j = 1, 2. For each such T we define a setβvalued map G~T~ from βοΈ(βοΈ~1~, βοΈ~2~) into the set of closed affine subspaces of βοΈ(βοΈ~1~^β²^, βοΈ~2~^β²^) by
G ~T~ (K ) = {K ^β²^ β βοΈ(βοΈ~1~^β²^, βοΈ~2~^β²^) : T ~21~ + T ~22~K = K ^β²^(T ~11~ + T ~12~K )} .
The map G~T~ 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 G~T~ is defined on some open ball of the space βοΈ(βοΈ~1~, βοΈ~2~).
Linear fractional relations are natural generalizations of linear fractional transformations studied by M. G. Krein and Yu. L. Ε muljan (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. (Β© 2006 WILEYβVCH Verlag GmbH & Co. KGaA, Weinheim)