Fusion-frames,which are special cases of g-frames in Hilbert space,share many similar prop-erties with g-frames,but it does not mean that all properties are similar.On the basis of the existing research,this paper uses the operator theory to discuss the equivalent characterization of fusion-Besselian frames,and a counter example is given to show that the conclusion is not valid without the finite dimen-sion space.Furthermore,the operator characterization of fusion-Besselian frames is given.Combining the operator characterization of fusion-Besselian frames with the first counter example,it is shown that the conditions and scope of application should be concerned.Then we discuss the relations among near fusion-Riesz bases,near Riesz bases and fusion-Besselian frames.Finally,the operator perturbations of the fusion-Besselian frames and the near fusion-Riesz bases are discussed,supplementing the research on operator perturbations.
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