G-frames And G-unitary Systems

The theory of frames plays a fundamental role in the signal processing, data compression, image processing and sample theory and so on. These applications promote the development of the theory of frames.In this thesis, based on the g-frame theory in Hilbert space introduced by Professor Sun Wenchang, the transformation and the dilation for the g-frames are investigated and some properties similar to the classical frame theory are obtained. The notions of the g-unitary system, g-wandering vectors and local commutant in Hilbert space are introduced and the dilation for g-unitary system is obtained. Then the notion of g-frame in Hilbert C*-module is introduced. Since there is no orthonormal basis in Hilbert C*-module in general, we dilate the g-frames to g-Hilbert base. The Hilbert C*-modules whose underlying C*-algebra is a compact operator algebra are called Hilbert K -module. There is standard orthogonal basis in Hilbert K -module. At the end of this thesis, the perturbation conditions about g-frames in Hilbert K -module are obtained.