Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các kép của một nhà điều hành dưới nhiệt độ. | J. Operator Theory 5 1981 195-211 Copyright by INCREST 1981 THE DUAL OF A SUBNORMAL OPERATOR JOHN B. CONWAY Let s be a pure subnormal operator on a Hilbert space that is s is a subnormal operator with no normal direct summand . If N is the minimal normal extension of s on the Hilbert space Jf containing then the dual of S is the restriction of 2V to X This paper studies some of the properties of this dual as well as those subnormal operators that are unitarily equivalent to their dual. The reader can consult 3 as well as the material and bibliography of 6 for the properties of subnormal operators. The theory of subnormal operators is receiving increased attention especially since it has been proved that they have nontrivial invariant subspaces 5 and indeed are reflexive 12 One of the difficulties with the study of subnormality is the lack of symmetry. The usual symmetry for operator theory involving the adjoint of the operator is completely absent. In fact if both s and s are subnormal then 5 is a normal operator. It is hoped that the concept of the dual of a subnormal operator will help to fill this void of symmetry. If G is an open subset of the plane and G - c then the analogue for f of the adjoint of an operator is given by complex conjugation. That is the adjoint of the function f is the function f G - c defined by z z . If f is analytic then is analytic iff f is constant on each component of G. Thus for spaces of analytic functions complex conjugation is an inappropriate form of symmetry. The appropriate form of symmetry for analytic functions is to define z z on the open set G z z e G . The dual of a subnormal operator is analogous to the function associated with an analytic function . Indeed there are results that relate the dual of a subnormal operator and the functions . See below. In Section 1 some of the elementary properties of the dual are established. Specifically if T is the dual of the pure subnormal operator s then s is the dual of T ữ T 2 2 6 ơ S
