tailieunhanh - Báo cáo toán học: "Redheffer products and the lifting of contractions on Hilbert space "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Redheffer sản phẩm và nâng bóp trên không gian Hilbert. | J. OPERATOR THEORY 11 1984 193- 196 Copyright by INCREST 1984 REDHEFFER PRODUCTS AND THE LIFTING OF CONTRACTIONS ON HILBERT SPACE CIPRIAN FOIAS and ARTHUR E. FRAZHO I t is the purpose of this note to show how Redheffer products 3 can be used to obtain a simple proof of the main result in 1 . In this paper all spaces are Hilbert spaces and all operators are bounded. We follow the standard notation in 2 5 . We begin by recalling some properties of Redhelfer products. Throughout 1 A B c D_ A B1 -Cl A. A are bounded operators mapping into S ty respec- tively where S ùlỉx and if 1. It is also assumed that the range of Ci and DrB C and are contained in 1 DjA 3 Z respectively. The Redheffer product is defined by 2 M La o L A1 B Ợ- - CỰ-DyA -1 BjAỰ - AT 1 DịB BjB Cự - D-lAY D-lB D where the inverse is the pseudo-inverse If T ly X then X is the unique element orthogonal to ker T such that Tx y. The Redheffer product exists whenever the entries in 2 exist as unique bounded operators coinciding with the linear transformations indicated by the entries. Redheffer did not use pseudo-inverses in his original paper 3 . However his results extend to our setting. Consider the system 3 A B ix z r A B 1 _ Ị and _ _ I I c D u U1 Cl Aj W17 z ur and X yỵ where u X y z wu Xj y1 and Zỵ are elements in the appropriate spaces. The Red-heffer product 2 is obtained by solving 3 in the following way M L1O M mM y u u 4 194 CIPRIAN FOIA and ARTHUR I . FRAZHO The operator J is defined by 5 J - 0 I I 0 where I is the identity operator on the appropriate space. J acts like the identity for the Redheffer product. To be precise L --L J J o L. If 7 is a contraction mapping into. u then Z 7 is the positive square root of I T ỈT and Í T Dyỹ í. We are ready for Lemma 1. Assume that M LjcZ where Lỵ and L in 1 are contractions. Then M is a contraction and 6 M ll2 ĩDLẠxy M1 ii2 dl x u i2 where X Uy are obtained by 3 . Proof. The proof is similar to some of the results in 3 For completeness it is given. .