Methods of Kreĭn Space Operator Theory

The author was originally led to Krĕın space operator theory by a problem of L. de Branges concerning the coefficients of univalent functions. The particular question was resolved in the negative, but the operator methods used to show this are related to other areas which remain currently active, such as the study of generalized Schur and Nevanlinna functions. The methods are of a general nature and based on familiar Hilbert space concepts, including contraction operators, their dilations, and reproducing kernel spaces. Today the Krĕın space counterparts of many of these ideas are complete to a high degree. As always, there are difficulties and new issues in the indefinite theory. For example, it turns out that uniqueness questions play a more important role in the indefinite theory than in the definite case. In this paper we survey some old and recent results in these areas, with an aim to show that tools which have found wide applicability in Hilbert space problems are also available in Krĕın space operator theory.