TAILIEUCHUNG - Báo cáo toán học: "Ultraweakly closed operator algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đại số nhà điều hành đóng Ultraweakly. | J. OPERATOR THEORY 2 1979 49-61 Copyright by INCREST 1979 ULTRA WEAKLY CLOSED OPERATOR ALGEBRAS CONSTANTIN APOSTOL The Invariant Subspace Problem remains unsolved since more than 30 years. The question to be answered is the following Does every bounded linear operator T acting in a separable complex Hilbert space have a proper invariant subspace Scott Brown 3 answered yes this question in case T is subnormal . the restriction to an invariant subspace of a normal operator . The way he solved the problem is beautiful but the most remarkable fact is that his techniques are strong enough for significant generalizations as well as for the study of the ultraweakly closed algebra generated by some operators. We already know three generalizations belonging to J. Agler 1 to s. Brown B. Chevreau and c. Pearcy 4 and to J. Stampfli 14 Given T assume that we have f T ứsup U l Aff f where a 1 is fixed and f is any rational function with poles outside ơ T . Then yes holds again as shown by J. Agler for a 1 and J. Stampfli for a 1. The Brown-Chevreau-Pearcy s theorem solves the problem for contractions T for which ơ T n Ấ I A 1 is dominating in the sense of Rubel and Shields see the Remark after Theorem . Along the same lines is our Theorem . below a generalization of Brown-Chevreau-Pearcy s result. The spectrum of our contraction T will include the unit circumference and the essential norm of the resolvent will grow quickly enough in the unit disk near ơ T . In the sequel we shall use the following notation H an infinite-dimensional complex Hilbert space Bh the open unit ball of H .y H the algebra of all bounded linear operators acting in H S tHy. the set of all trace-class operators acting in H s the adjoint of a unilateral shift acting in H the ultraweakly closed algebra generated by all polynomials in T where T belongs to f H D the interior of the unit disk. ãẽõ the closure of the absolutely convex hull. 50 CONSTANTIN APOSTOL Consider the bilinear functional on X W A

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