四元数Hilbert空间中Riesz基的刻画*

四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论, 在四元数Hilbert空间中引入了Riesz基的概念, 在此基础上刻画了Riesz基,给出了它们的一些等价条件; 特别地, 得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列; 并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系.

Characterizations of Riesz Bases in Quaternionic Hilbert Spaces

Quaternionic Hilbert spaces play an important role in applied physical sciences especially in quantum physics. This paper addresses the frame theory in quaternionic Hilbert spaces. The authors introduce the notion of Riesz bases in quaternionic Hilbert spaces.Then they characterize Riesz bases in this setting, present their some equivalent conditions;particularly, they obtain that a sequence in quaternionic Hilbert spaces is a Riesz basis if and if only it is a complete Bessel sequence with biorthogonal sequence which is also a complete Bessel sequence; and further prove that the completeness of one (any one) of the biorthogonal sequences can be removed from the characterization. Some examples are given to illustrate the relations between biorthogonality, completeness and Bessel property.