N Subspaces

Abstract

It is a well-known fact (cf., for instance Lemma 7.3.1 of [8] , and also [2] and [4] ) that if M and N are closed subspaces of a finite-dimensional Hilbert space, and if M and N are in ‘generic’ position (i.e., any two of the four subspaces M, M ⊥ , N, N ⊥ have trivial intersection), then N is the graph of a linear isomorphism of M onto M ⊥ . To be sure, there exist infinite-dimensional versions of this, where one must allow for unbounded operators in case the ‘gap’ between M and N is zero, in the sense of Kato [7]. (There is an extensive literature on pairs of subspaces, [2], [3], [4], [6] and [7], to cite a few; for a fairly extensive bibliography, see [3] .) This paper addresses itself to the case of n (2 ≦ n < ∞) subspaces.