高阶导算子为厄米特算子的几个充要条件

SOME NECESSARY AND SUFFICIENT CONDITIONS ON THE DERIVATION IS HERMITIAN OPERATOR

在这篇文章里,我们证明了下列条件是等价的:(i)δ_m^((k))(A)是厄米特的;(ii)(Av,v)~k∈R,(?)v∈V;(iii)A=cH,H=H~*且c^k∈R;(iv)当m<n(=dimV)时,W~⊥(δ_m^((k))(A))(?)R。

Let L(V) denote the set of all linear operators on an n-dimensional uni- tary space V.Let (?)V be the mth tenor product space of V.For A(?)L(V), δ_m^((k))(A) is kth derivation of (?)A.For L(?)L((?)V),W~⊥(L) is orthonomal nu- merical range of L.This paper proves that the following conditions are equiv- alent:(i)δ_m^((k))(A) is hermitian;(ii)(Av,v)~k(?)R,(?)v(?)V;(iii)A=cH,H=H~* and c^k(?)R;(iv)W~⊥(δ_m^((k))(A))(?)R if m<n.