tailieunhanh - Báo cáo toán học: "Bi-invariant subspaces of weak contractions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Bi-subspaces bất biến của các cơn co thắt yếu. | J. OPERATOR THEORY 1 1979 261-272 Copyright by INCREST 1979 BI-INVARIANT SUBSPACES OF WEAK CONTRACTIONS PEI YUAN wu 1. INTRODUCTION For a bounded linear operator T acting on a complex separable Hilbert space H let Alg T T and r denote the weakly closed algebra generated by T and I the double commutant and the commutant of T respectively. A subspace K of H is said to be bi-invariant resp. hyperinvariant for 7 if A is invariant for every operator in T resp. T . LetLat T Lat Tand Hyperlat Tdenote the lattices of invariant subspaces bi-invariant subspaces and hyperinvariant subspaces of T respectively. The following trivial relations hold Alg T T s T and Lat T 3 Lat T Hyperlat T. For various classes of operators among which are normal operators and operators acting on a finite dimensional space the elements of Lat T have been completely determined cf. 4 . In particular if T satisfies the double commutant property that is if Alg T T then Lat Tcoincides with Lat T. The purpose of the present paper is to study Lat 7 for completely non-unitary . weak contractions with finite defect indices. Before in a series of papers 13 14 and 15 we investigated the elements of Hyperlat T for such operators. These were preceded by the work of Sickler 5 . We gave specific descriptions of the elements of Hyperlat T and showed that Hyperlat T is preserved as a lattice under quasi-similarities of this type of operator. In this paper we extend some of these results to Lat T. As before we shall develop the theory in two stages first for cu contractions and then for weak contractions. In Section 2 we fix the notation and terminology and briefly review some basic results needed in the later work. In Section 3 we consider . cn contractions with finite defect indices. Specific descriptions of the elements in Lat T are given in Theorem . As a corollary we show that Lat T is also preserved under quasi-similarities Corollary . The former result is then extended to . weak .