NEAR SUBNORMAL OPERATORS AND SUBNORMAL OPERATORS

It has ever been regarded as non-trivial to answer Halmos’ problem 160. Now there is a simple example to answer the problem. It, however, is in isolation, from which we can. not find any class of new operators between subnormal and hyponormal operators. In this paper, a class of near subnormal operators is introduced. For weight shift "operators, it is convenient to verify the near subnormality. Then, a new sufficient and necessary condition for subnormal operators is given. It is not very difficult to verify subnormality of the preceding operators. Thereby, we present two interesting weight shifts, namely, one is hyponormal but not near subnormal, the other is near subnormal but not subnormal. This is a further answer to Halmos’ problem 160.