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Báo cáo toán học: "Completing matrix contractions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hoàn thành các cơn co thắt ma trận. | J. OPERATOR THEORY 7 1982 179 189 Copyright by INCREST 1982 COMPLETING MATRIX CONTRACTIONS Gr. ARSENE and A. GHEONDEA The aim of this note is to describe all solutions of the following problem Let H Hr@ H2 K Kỵ K2 be Hilbert spaces and A e ỈTịHỵ Kj B e H Á1 ce j2 such that A A B and ÍA B be contractions. Find all X e y H2 Kz such that A I I be a con- vc x traction. Applications of this labelling to dual pairs of subspaces in a Krein space to dual pairs of accretive operators as well as to extensions of positive unbounded operators are given. Thanks are due to Zoia Ceausescu for helpful discussions on the subject of this paper. 1. MAIN THEOREM We consider complex Hilbert spaces and we denote by T H K the set of all linear bounded operators from the Hilbert space H into the Hilbert space K. For K i.e. T is a contraction that is T 1 let DT I - T T 1 2 and DT H be the defect operator respectively the defect space of T. We shall use the following result which is proved in this form in 5 Lemma 1.1.2. Lemma 1.1. Let H and K be Hilbert spaces and suppose that H Hi H2 and TjS Hỵ K . Then the formula 1.1 T T1 DjJ.T 180 Gr. ARSENE and A. GHEONDEA establishes a one-to-one correspondence between all T 6 4 1 F K such that TịH-i Tỵ and all r e Moreover the operators 1.2 Z Ti T - z QT1 zr ữT Z Dt Dr - AW P J AW 1 j.CJXzj. n Í 1.3 are unitary operators. Z iTỵ T Z ữj - ữT zAjL J.D For Ho c H PfỊữ denotes the orthogonal projection of H onto Ho. Let US note that the correspondence described in 1.1 first appeared in 7 and that formulas similar to 1.2 and 1.3 were used in 6 and 3 In the proof of 1.1 a well known result on the factorization is used namely A A B B if and only if A CB where c is a contraction see for example 9 Theorem 1 . For the reader s convenience we repeat here the proof from 5 for the formula 1.2 . We will prove that 1-2 I Drjh VPr i. for every hỵ e Hỵ and that 1.2 IIA aL PD h. . for every h2e H where we put p - Pgr which will imply 1.2 . We have Whir 11 1 r - w 2 I.2