幂等算子线性组合的幂等性

Idempotency of Linear Combinations of Two Idempotent Operators

设P与Q是Hilbert空间中的两个不同的幂等算子.本文主要刻画了幂等算子P与Q的线性组合仍是幂等算子的充要条件,从而推广了Baksalary与Baksalary (2000)的结论.值得指出的是,我们通过严密的推理发现,其定理的条件P_1P_2≠P_2P_1是非必要的.

Let P and Q be two different idempotents on a Hilbert space. In this note, a complete solution is established to the problem of characterizing all situations, where a linear combination of P and Q is also an idempotent. Besides, we pointed out that the condition PIP2 ≠P2P1 of the statement （b） of the theorem in Baksalary and Baksalarv （2000） can be deleted.