保测变换的特征算子

Eigenoperators of Measure Preserving Transformations

设（Ｓ，，ｍ）是一个可分概率空间，Ｅ是复的可分Ｂａｎａｃｈ空间，ｈ：Ｓ→Ｓ是（Ｓ，，ｍ）上的保测变换，Ｘ：Ｓ→Ｅ是非零的Ｂｏｒｅｌ可测函数，Ｔ是Ｅ上的有界可逆线性算子．假定Ｘ（－ｈ（·））＝ＴＸ（·），ａ．ｅ〔ｍ〕．就称Ｔ是ｈ的特征算子，Ｘ是ｈ的特征函数．证明了若Ｅ是ｔｙｐｅ－２空间，那么Ｔ表示为保测变换ｈ的特征算子且ｈ的特征函数为平方可积的充要条件是存在正定对称算子Ｒ：Ｅ→Ｅ。

If E is a complex separable type-2 Banach space,T is an invertible linear operator on E , then[KG*3/5]the following areequivalent: (a) T satifies the eigenoperator equation X(h(·))=TX(·),a.e m where h is a measure preserving transformation in a separable probability space (S,,m) and X:S→E is not identically 0 square integrable function.(b) there exist a positive symmetric operator R:E →E: such that TRT =R and the square root of R is 2 absolutely summing.