亚正规算子的性质

The Properties of Hyponormal Operators

研究了亚正规算子A∈B(H)的性质,给出了A∈B(H)的约化子空间的充分条件,并且讨论了其与正规算子的关系.得到了定理1设A∈B(H)是一个亚正规算子,则以下结论成立:(1)若0<p≤1,则A为p-亚正规算子.(2)对于每一个正整数n,An∈B(H)为1n-亚正规算子.(3)若0<p≤1,则|A|2p≥|A*|2p.(4)对每一个正整数n,有‖An‖=‖A‖n.定理2设A∈B(H)是一个亚正规算子,W是A的不变子空间,且A在W上的限制A|W为正规算子,则W是A的约化子空间.定理3设A∈B(H)是一个亚正规算子,且A*下有界,则A是可逆的,且A-1也为亚正规算子.

The properties of hyponormal operators are studied. Firstly, the sufficient condition of the re duced space of hyponomal operator A is given; secondly, the relation of the hyponormal operator to the normal operator is discussed.Theorem 1 Let A∈B（H）. A is a hyponormal operator, then the followings hold： （1） If 0〈p≤1, then A is a p-hyponormal operator.（2） For every positive integer n, A^n∈B（H） is a 1/n-hyponormal operator. （3） If 0〈p≤1, then ｜A｜^2p≥｜A^＊｜^2p （4） For every positive integer n, we have ｜｜A^n｜｜=｜｜A｜｜^n. Theorem 2 Let A∈B（H）. A is a hyponormal operator,W is an invariant subspaces of A and A｜W is normal, then W is a reduced subspace of A. Theorem 3 Let A ∈ B（H）. A is a hyponormal operator, A^＊ is bounded below, then A is invertible and A^-1 is hyponormal.