Norm Preserving Extensions of Linear Transformations on Hilbert Spaces

Introduction. Let 77 be a Hubert space and let D be a closed proper subspace of 77. Let 70 be a linear contraction on D to 77. The problem of characterizing the contractions on all of 77 which extend J0 is directly related to the extension problems for unbounded transformations posed and treated by M. G. Krein [2] and R. S. Phillips [3]. In §1 of this paper we establish the following solution of this problem: Let P be the orthogonal projection of 77 on D and Ji — JoP. The contractions on all of 77 which extend 70 are precisely the transformations of the form Ji + (I — JiJ*)ll2B, where B is any contraction which annihilates D. §2 is a short outline of the relationship of this result to the extension problem for dissipative transformations treated by Phillips in [3] and [4]. §3 contains some remarks concerning the case where Jo is symmetric and one asks for the selfadjoint contractions which extend Jo, which is essentially the situation treated by Krein [2]. In particular, we obtain the Krein extension of a symmetric contraction to a selfadjoint contraction by, we feel, a less elegant but more illuminating argument than Krein's original one.