tailieunhanh - Báo cáo toán học: "Extensions of normal operators and hyperinvariant subspaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Phần mở rộng của các nhà khai thác bình thường và subspaces hyperinvariant. | J. OPERATOR THEORY 3 1980 203-211 Copyright by INCREST 1980 EXTENSIONS OF NORMAL OPERATORS AND HYPERINVARIANT SUBSPACES H. w. KIM and c. M. PEARCY 1. INTRODUCTION Let be a separable infinite dimensional complex Hilbert space and Jet S .J denote the algebra of all bounded linear operators on 2 Í. If TV is a nonscalar normal operator in then the spectral theorem guarantees that TV has a generous supply of nontrivial hyperinvariant subspaces. Recall that a subspace 11 of is called a nontrivial hyperinvariant subspace for an operator T in ỈỀựe if 0 7 11 7Ế and r ll lc for every T in that commutes with T. For this reason classes of operators that are intimately associated with normal operators have long been favorite objects of study with a view to establishing the existence of nontrivial invariant and hyperinvariant subspaces. In this connection the classes of n-normal operators and subnormal operators come quickly to mind. Recall that an n-normal operator may be defined as an n X n operator matrix whose entries are mutually commuting normal operators and a subnormal operator is the restriction of a normal operator to an invariant subspace. In 5 it was shown that every nonscalar n-normal operator has nontrivial hyperinvariant subspaces cf. also 9 p. 76 and 11 . The corresponding problem for subnormal operators remains unsolved cf. 8 but in the pioneering paper 1 Scott Brown recently proved that every subnormal operator in .if j does have nontrivial invariant subspaces. The purpose of this note is to make a beginning on the hyperinvariant subspace problem for another class of operators closely related to the normal operators namely the class of extensions of normal operators. Recall that an operator T is an extension of a normal operator if T restricted to some nontrivial invariant subspace of r is normal. In what follows we call such operators extnormal operators. Thus by definition extnormal operators have nontrivial invariant subspaces and it is only the question of