B(H)表示定义在希尔伯特空间H上的所有有界线性算子的全体。对于A∈B(H),其中σ(A)和W(A)分别表示算子A的谱和数值域,N表示自然数集。关于算子A的n(n∈N)次方根,本文的主要结果是:(1)若σ(A)∩(-∞,0]=φ,则A有惟一的n次方根B∈B(H)且σ(B)(?)~(2/n)~o;(2)若(?)∩(-∞,0]=φ,则A有惟一的n次方根B∈B(H)且(?)(2/n)~o这里,S_(1/n)={λ∈C‖argλ|≤(1/2n)π}且S_(1/n)~o表示集合S+(1/n)的内部。

For A∈B(H), let σ(A) and W(A) denote the spectrum of A and the numerical range of A respectively. In the present paper we show two important results: one is that if σ(A)∩(-∞, 0]=φ then A has a unique n-th root B∈B(H) with σ(B)(?), and the other is that if (?)∩(-∞, 0]=φ then A has a unique n-th root B∈B(H) with (?), where S_(1/n)={λ∈C‖argλ|≤1/2nπ} and S_(1/n)~0 denotes the interior of the set S_(1/n).