Banach空间上有界可逆线性变换逆变换的结构

The Structure of Inverse Transformation for Linear Invertible Bounded Transformation on Banach Space

Fillmore在[1]中得到一个定理:设A,T是Banach空间X上的线性变换,A有界,若Lat(A) Lat(T)且AT=TA,则T是A的多项式.在本文里,以此作为引理,讨论了Banach空间上可逆线性变换A在什么情况下,A-1可表示为A的多项式.本文最主要的结论是定理3.4:设X是Banach空间,A是X上的有界线性变换,且可逆,则A-1是A的多项式当且仅当A-1是A的局部多项式.

It was shown by Fillmore in [1] that linear transformations A and T on a Banach space, with A bounded. If Lat(A)Lat(T) under what conditions and T commutes with A, then T is a polynominal in A. In this paper, we discuss the problem: discuss the problem that A-1 is a polynominal in A when A is a linear invertible transformation on a Banach space. The main result is that if A is a linear transformation on a Banach space with A bounded and invertible, then A-1 is a polynominal in A if and only if A-1 is locally a polynominal in A.