The Stability Of Some Frames In Hilbert Space

The frame theory is a new research direction after wavelet analysis. In1952,Duffin and Schaeffer defined frames in Hilbert space when they studied thenonharmonic Fourier series, the frame theory is widely used in both cynosural fieldtheory and application area. After a period of development, some studiers have givenother definitions more general than frames, such as pseudo-frames, oblique frames,outer frames, frames of subspaces and g-frames. The g-frames were defined byWenchang Sun in2005. As the extension of frames, g-frames have several differenceswith frames. In order to find these differences, both the studiers in foreign countiesand in inland spread a lot of researches in its characters. Among all the characters, thestability of g-frames plays an important part in the research of g-frames. It is helpfulto know about the structures and characters of g-frames.In the first part of the paper, we apply operator theory to the study of g-frames. Atfirst, we combine the thought of operator disturbance and matrix disturbance, useoperator matrix to perturb g-frames, and the operator matrix is the matrix whoseelements are operators. When we let the elements of operator matrix as constants, theoperator matrix disturbance is the normal matrix disturbance. Specially, let thenon-diagonal elements as zero, we get another form of disturbance, which is operatorsequence disturbance. Afterwards we prove that g-Riesz frames and g-framesequences have similar results. Further we get the result that there must be an operatormatrix between two g-frames.In the second part, we study the relationship between g-Riesz basis, g-Besselianframes, near-g-Riesz basis, g-Riesz frames and the frames in their correspondingsubspaces. From the obtained result, we can get the relationship between g-Besselianframes, near-g-Riesz basis and the orthonormal basis in their corresponding subspaces,which have been discussed. Then we discuss the stability of these g-frames underperturb of operator sequences.