g-框架与g-标准正交基的一些性质

Some Properties of g-frame and g-orthonormal Basis

g-框架作为Hilbert空间中传统框架的推广,具有很多良好的性质.首先运用算子理论的方法研究了g-框架在预框架算子、合成算子的作用下的稳定性;然后探究了紧g-框架在正交补空间上的性质,并用Schmidt正交化法证明了无冗余紧g-框架构成Hilbert空间H中正交基的必要条件;最后研究了g-Bessel序列构成g-标准正交基的充要条件以及g-标准正交基在单射算子扰动下g-双正交序列的存在性.

As a generalization of the traditional frame in Hilbert space,g-frame has many good properties.The stability of g-frame under the action of preframe operator and composition operator is studied by means of op-erator theory,and then the properties of tight g-frame on orthogonal complement are investigated,the necessary and suffcient condition of g-Bessel sequences forming g-Orthonormal basis in Hilbert space H is proved by Schmidt orthogonalization,and the existence of G-biorthogonal sequences for g-Orthonomal basis perturbed by a single-shot operator.