关于p-弱亚正规算子的一个注记

A Note on p-ω-Hyponormal Operators

对0<p<1,本文引进了p-弱亚正规算子的概念,这类算子包含所有的弱亚正规算子．该类算子的若干性质被得到．首先,它与弱亚正规算子有许多类似的性质, 如它的谱半径与其范数相等、联合点谱与点谱的非零点相同等等;其次,指出对某些特殊的p-弱亚正规算子其平方仍是p-弱亚正规算子,而对一般的p-弱亚正规算子此结论未必成立．最后,得出在有限维空间中除了正规算子外没有其他的p-弱亚正规算子．

For 0 〈 p 〈 1, the class of p-ω-hyponormal operators is introduced. This class contains all ω-hyponormal operators. Certain properties of this class of operators are obtained. First, many properties that the ω-hyponormal operators possess are shown to hold for the p-ω-hyponormal operators; for example, if T is a p-ω-hyponormal operator, then its spectral radius and norm are identical, and the nonzero points of its joint point spectum and point spectum are identical. Secondly, for some special p-ω- hyponormal operators, we also prove that their squares are p-ω-hyponormal operators; but in general, it is not true for all p-ω-hyponormal operators. Lastty, we show that there is no p-ω-hyponormal operator in an n-dimensional space unless it is a normal operator.