Hilbert空间中有效序列的刻画与扰动

Characterizations and Perturbations of Effective Sequences in a Hilbert Space

本文研究了Hilbert空间中的有效序列,通过引入序列的伴随序列,给出了有效序列的一个等价刻画,接着讨论了保持向量序列有效性的线性算子的性质,证明了一个线性算子将任意有效序列映射为有效序列当且仅当它是酉算子;最后,给出了正规正交基中添加一个单位向量后所得到的序列是有效序列的一个充要条件.

In this paper, the effective sequences in a Hilbert space are discussed and by introducing the adjoint sequence of a sequence, an equivalent characterization of an effective sequence is presented. Next, some properties of linear operators preserving the effectiveness of effective sequences are discussed. It is proved that a linear operator maps every effective sequence as an effective sequence if and only if it is a unitary. Finally, we give a necessary and sufficient condition for a sequence obtained by adding a unit vector into an orthonormal basis to be an effective sequence.