Frames In Hilbert K-modules

The theory of frames plays a fundamental role in the signal and image processing,data compression, and sampling theory and more,as well as being a fruitful area for the research of abstract mathematics. The study of frame theory becomes very important. For example,the study of disjointness of frames is valuable in judging the communication of informations and signals.In this paper, we mainly consider frames in modules on compact operator algebras which is called K-module. Because K-module frames can not be dilated, we give the definitions of frame transform, frame operator, canonical dual frame and alternate dual frame of a frame in Hilbert K-module again, and obtain some basic properties. We study the disjointness of frames and stabilities of Bessel sequence, Riesz bases, Hilbert bases and frames in Hilbert K-modules, and we have the (strong) disjointness of frames related to the range of their frame transforms. When we consider Hilbert C~*-subalgebras as Hilbert K-modules on itself, we obtain two important results by the theories of index for C~*-subalgebras.We introduce the concept of unitary system acting on Hilbert K-modules, frame vectors, and local commutant and so on, and parameterize some frame vectors for unitary system acting on Hilbert K-modules. We solve the approximation problem of mult-frame vectors, and we use normalized mult-frame vectors to approximate general mult-frame vectors.Finally, we consider the unitary system acting on Hilbert K-modules as a unitary group, and define the frame representation for unitary group on Hilbert K-modules. We introduce a~*- homomorphism between unitary system U and U (Μ), where U (Μ) is the group composed of unitary operators acting on Hilbert K-modulesΜ.We mainly prove that the frame multiplicity for group representation is finite.