Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Phổ điểm approximative của một nhà điều hành dưới nhiệt độ. | J. OPERATOR THEORY 5 1981 119-126 Copyright by INCREST 1981 THE APPROXIMATE POINT SPECTRUM OF A SUBNORMAL OPERATOR WILLIAM w. HASTINGS Let 5 be a pure subnormal operator on a separable Hilbert space .5 . By pure we mean that no subspace of reduces s to a normal operator. Let Ko ơ S and let Kỵ ơa S the spectrum and approximate point spectrum respectively of s. Then we must have 1 dKữ Kỵ c Ko. 2 If A is an open disk for which A n Ko 0 then C d-n Xo n Xo . For a compact set K in the plane C K is the space of continuous complex-valued functions on K with the sup-norm and R K is the closure in C X of the set of rational functions with poles off K. Condition 1 is a well-known result of Halmos 5 and Bram 2 and 2 is the necessary and sufficient condition found by Cllancey and Putnam 3 in order that the compact set Ko be the spectrum of a pure subnormal operator. In this paper we investigate which pairs A o Kr of compact sets arise in this way as the spectrum and approximate point spectrum of a pure subnormal operator. In general the necessary conditions 1 and 2 are sufficient. We will require however that the subnormal operator 5 have additional properties. This yields results with implications for example to the study of -approximation by rational functions. An operator A .X - is effectially rationally cyclic cf. 1 if there exists a vector e J F such that the set yerf p A y q A x0 for some polynomials p and q with p A 0 is dense in Berger and Shaw 1 have shown that an effectually rationally cyclic hyponormal operator s is essentially normal 5 s ss is compact . If K is a compact subset of the plane then a hole of K is a bounded component of the com 120 WILLIAM w. HASTINGS plement. The polynomial convex hull of K denoted K is the union of K and all of its holes. The following results are proved in this paper. Theorem 1. Let Kữ and Kỵ be compact subsets of the plane which satisfy 1 and 2 . Then there exists a pure subnormal operator s such that ơ S Kữ and cfS Kv If Kr has no
