Sur les variétés linéaires dans les espaces à métrique indéfinie et les décompositions orthogonales des espaces. (On linear manifolds in indefinite inner product spaces and orthogonal decomposition of spaces) (Q1114118)

This paper suffers from the drawback that the list of references is not included. However, to quote from an editorial note, ``...the paper can be read without this list of references (and some of them can be guessed at).'' The first section deals with properties of decomposable indefinite inner product spaces with complete negative part. Some properties of semidefinite linear manifolds that decompose the space are studied. In the second section, a TU-space is defined as a space equipped with a symmetric sesquilinear form, which can be decomposed into an orthogonal direct sum of a positive and a negative linear manifold, at least one of which is complete with respect to the restriction of the form. The principal result of this section is the theorem: If a prehilbert \(J_ 2\)-space \({\mathcal N}\) is a TU-space, then every decomposition of \({\mathcal N}\) into an orthogonal direct sum of a positive manifold \({\mathcal L}\) and a negative manifold \({\mathcal L}'\) satisfies certain equivalent conditions, which include: (a) the decomposition gives rise to bounded projections; (b) \({\mathcal L}\) and \({\mathcal L}'\) are uniformly definite; (c) the norm on \({\mathcal N}\) is equivalent to the norm induced from the sesquilinear form by the decomposition. Examples are given to show that the TU-space condition can not be dropped.