Operator-valued Frame Generators For Unitary Systems And Group-like Unitary Systems

In this thesis, we mainly investigate the operator-valued frames on Hilbert spaces. Especially,we are focused our attention on the study of the properties and conclusions of operator-valuedframe generators for unitary systems and group-like unitary systems.Firstly, motivated by operator-valued frames and unitray systems on Hilbert spaces, weintroduce the concept of operator-valued frame generators for unitary systems. Some properties ofoperator-valued frame generators, for example, left similarity, right similarity, orthogonality, dualityand so on, are discussed. Combining the the commutants of unitary systems, we study theproperties of the analysis operators, frame operators and frame projections which are determined byframe generators. In particular, we characterize the dilation and the disjointness ofoperator-valued frame generators, for a special class of unitary systems which is determined by theunitary representation of a countable discrete group on a Hilbert space. In addition, a necessary andsufficient condition for a bounded linear operator to become a Parseval operator-valued framegenerator for a certain unitary system is given.Secondly, taking the projective unitary representation of a countable discrete group intoaccount, we extend the concept of operator-valued frame generators from unitary systems togroup-like unitary systems, and describe some properties of them. Also, we obtain a necessary andsufficient condition for group-like unitary systems to have operator-valued frame generators.