投影算子组的联合谱

本文研究Hilbert空间H上投影算子组的联合谱.首先通过计算给出正则投影对的联合谱,进而给出一般的投影算子对的联合谱.本文还对两个投影算子的和与差的可逆性给出一些等价刻画.特别地,当P和Q为正则投影对时,本文通过计算算子组I, P, Q]的联合谱来给出σ(P+Q)和σ(P-Q)的具体刻画.反过来,本文证明两类具有特定形式的复数集分别是Hilbert空间上正则投影对的和与差的谱.本文也给出一般的投影算子对的和与差的谱.最后,本文计算特定条件下的3个投影算子组P, Q, R]的联合谱.

The joint spectrum of a tuple of projections

We study the joint spectrum of projections in a Hilbert space, and calculate the joint spectrum of a pair of projections. In particular, by calculating the joint spectrum for a tuple I, P, Q] in which I is the identity and P, Q is a regular pair of projections, we get specific characterizations of the spectrums of P + Q and P-Q,respectively. Conversely, we prove that two closed subsets of C in particular forms are respectively the spectrums of the sum and the difference of a regular pair of projections. We also calculate the joint spectrum of a 3-tuple of projections P, Q, R] for P, Q and R satisfying some specific conditions.