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respect to T such that in one of them T induces a unitary operator whereas the part of Tin the other is completely nonunitary. In this way, the unitary part of the operator acts in an isolated part of the HILBERT space and, consequently, it is not connected with the remaining part of the contraction. What concerns semiunitary operators the situation is completely different. We call an operator U serniunifary if U*U = 1 or U U * = 1. In the first case U is called isometric whereas in the latter case U is said to be coisometric. If, for example, a completely nonunitary contraction T : -, 4 induces an isometric operator on some (closed) invariant subspace I), E 4 then this means that V o : = & d~. b o T is a unilateral shift and the subspace bo does not necessarily reduce 7: It turns out that if T contains a unilateral shift V,, then this implies that its characteristic function O(z) has some defect. Analogously, if T* contains a unilateral shift then this induces another kind of defect related with O(z). This paper is aimed at a detailed investigation of the indicated connections.In Section I , geometric questions linked with maximal unilateral shifts contained in Tand T*, respectively, will be studied.In Section 2, defect functions will be introduced. Moreover, a description of all regular extensions of holomorphic contractive matrix functions in the unit disc will be given.Section 3 is dedicated to an interpretation of defect functions in the language of open systems.It is widely known that the role of bilateral shifts in the theory of contractive operators is thoroughly discussed in the monograph SZ

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Introduction [40]. It is remarkable that on the one hand, the problems treated in these monographs essentially coincide, whereas, on the other hand, the methods used there are almost completely different. This phenomenon reflects the many facets of the objects studied in SCHUR analysis. U p to now the cross connections between the various approaches to interpolation are only roughly uncovered. In order to get deeper insights into the nature of the structures in question, it seems to be fruitful to carry out a very detailed analysis of these cross connections. First attempts in this direction had been started in [28] The authors agree with V. E. KATSNELSON, who expressed in his lecture at the SCHUR Analysis Seminar, held in Leipzig in October 1989, the opinion that the treatment of more complicated interpolation problems will not be successful without combination of elements taken from various approaches. Only this way seems to ensure that essential features of the objects in question will not be neglected. A first remarkable step to confirm this point of view was done by KATSNELS~N/HEIFETS/JUDITSKI~ [44], who have