Cowen-Douglas算子的一些注记

取定Cowen-Douglas算子T∈_n(Ω)， 给出了其对应的复解析丛E_T的一类特殊截面， 进而引入Cowen-Douglas算子一类新的更易计算的酉不变量［Φ］. 在n≥2的情形, ［Φ］是n×n的复光滑函数值矩阵Φ(T)的对合等价类, 特别地, 在B₁(Ω)的情形, 其为实值函数. 在此基础上， 给出一类 Cowen-Douglas算子的分解惟一性. 证明了当一个Cowen-Douglas算子T满足D［Φ］>n²-2n+2时, T是Hilbert不可约的.

Some Remarks on Cowen-Douglas Operators

For a given Cowen-Douglas operator T∈B_n(Ω), a special kind of cross-sections of the corresponding complex bundle E_T is presented. A new unitary invariant ［Φ］ is introduced, which is the conjugate classes of the complex C^∞ function valued n×n matrix Φ(T) i n the case of n≥2 and a real function in the case of n=1. For a given matrix Φ(T)=(φ_{ij/sub>)∈［Φ］, let F=∨ⁿi,j=1{φij}. dim F is independent of the choice of Φ(T). Define D［Φ］=dim F. By a di scussion on the behavior of D［Φ］, we show the uniqueness of a special kind of Cowen-Douglas operators. Moreover, we have proved that if a Cowen-Douglas operator T∈Bn(Ω) satisfies D［Φ］≥n²-2n+2, then T must be Hilbert irreducible.}