Complementation in Krein Spaces

Abstract

A generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product. Complementation is a construction in the geometry of Hilbert space which was applied to the invariant subspace theory of contractive transformations in Hilbert space by James Rovnyak and the author [6]. The concept was later formalized by the author [3]. Continuous and contractive transformations in Krein spaces appear in the estimation theory of Riemann mapping functions [4]. It is therefore of interest to know whether a generalization of complementation theory applies in Krein spaces. Such a generalization is now obtained. The results are also of interest in the invariant subspace theory of continuous and contractive transformations in Krein spaces [5]. The vector spaces considered are taken over the complex numbers. A scalar product for a vector space )1 is a complex-valued function (a, b) of a and b in )1 which is linear, symmetric, and nondegenerate. Linearity means that the identity (aa + 3b, c) = a(a, c) + 3(b, c) holds for all elements a, b, and c of )1 when a and , are complex numbers. Symmetry means that the identity (b, a) = (a, b) holds for all elements a and b of )1. Nondegeneracy means that an element a of )1 is zero if the scalar product (a, b) is zero for every element b of )1. Every element b of )1 determines a linear functional bon )1 which is defined by b-a = (a, b) for every element a of )1. The weak topology of )1 is the weakest topology with respect to which bis a continuous linear functional on )1 for every element b of )1. The weak topology of )1 is a locally convex topology having the property that every continuous linear functional on )1 is of the form bfor an element b of )1. The element b is then unique. The antispace of a vector space with scalar product is the same vector space considered with the negative of the given scalar product. A fundamental example of a vector space with scalar product is a Hilbert space. A Krein space is a vector space with scalar product which is the orthogonal sum of a Hilbert space and the antispace of a Hilbert space. Received by the editors October 10, 1986 and, in revised form, February 23, 1987. The results of the paper were presented to the Department of Mathematics, Indiana and Purdue University in Indianapolis, on March 27, 1987, as the Ernest J. Johnson Colloquium. 1980 Mathematics Subject Classification (1985 Revision). Primary 46D05. Research supported by the National Science Foundation. ?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page