关于算子组的本质谱

ON ESSENTIAL SPECTRUM OF OPERATOR SYSTEMS

在单个算子理论中,我们知道:一个Babach空间X上的有界线性算子T为Fredholm的充分必要条件是它在X_q=l~∞(X)/P_c(X)上的诱导算子T_q为可逆。本文主要是把这一结果推广到算子组。作者证明了:有限级Banach空间复形(X^p,α~p)_(p=0)~n为Fredholm的充分必要条件是相应的复形(X_q^p,α_q^p)_(p=0)~n为正合的,这里α~p为有界的。进而得出:对于交换有界线性算子组T=(T_l,…,T^n),成立着Sp_e(T_1,…,T_n)=Sp(T_(1q),…,T_(nq))。最后利用以上结果讨论了一些算子组成为Fredholm的充分条件。

It is well know that a bounded operator T: X→Y, where X and Y are Banach spaces, is Fredholm if and only if the operator T_q : l~∞(X) / Pc(X)→l~∞(Y) / Pc(Y)induced by T is invertible, where l~∞(X) (or l~∞ (Y)) denotes the space of all bounded sequences of X (or Y) and Pc(X) (or Pc(Y)) denotes the space of all precompact sequences of X (or Y). The aim of this paper is to generalize this result to commuting operator systems. The author proves that a limited complex of Banach spaces (X^p, α~p)_p^n=0, where α~p is bounded for p=0,…, n-1, is Fredholm ff and only if the corresponding complex (X_q^p,α_q^p)_p^n=0 is exact, where X_q^p=l~∞(X^p)/Pc(X^p) and α_q^p; X_q^p→X_q^(p+1) is induced by α~p for p=0,…, n-1, and then we obtain that Sp_e(T_1,…,T_n)= Sp(T_(1q),…,T_(nq)) for T= (T_1,…,T_n), a commuting system of bounded linear operators on a Banach space. At last, making use of the above results, the author presents some Sufficient conditions under which a commuting system of bounded operators is Fredholm.