Dualizable Q-Frames, Q-Besselian Frames And Q-Riesz Frames In Banach Spaces

Because of putting forward the concepts of q-frame and p-Riesz basis, some properties offrames in Hilbert spaces have been directly extended to Banach spaces. On this foundation,we introduce the concepts of dualizable q-frame, q-Besselian frame and q-Riesz frame inBanach spaces. Using operator theory and method of functional analysis, we study theproperties of these three frames correspondingly. This paper consists of four chapters.Chapter 1 is about introduction and fundamental theory. Introducing the concepts ofanalysis operator and synthesis operator, we discuss the properties of q-frames and p-Rieszbases in Banach spaces. We obtain many similar conclusions to ones in Hilbert spaces.These conclusions lay a foundation for the later studies. Finally, we introduce the mainperturbation results on q-frames and p-Riesz bases and obtain a theorem about theperturbation of p-Riesz basis.Chapter 2 is about dualizable q-frame. First, we introduce the definition of dualizableq-frame in Banach spaces and give some necessary and sufficient conditions of dualizableq-frame. Next, we discuss the stability of dualizable q-frame under perturbation. Ourperturbation result is a generalization of the perturbation theorem in Hilbert spaces. Chapter 3 is about q-Besselian frame. To begin with, we get the one-to-onecorrespondence relation between a q-Besselian frame and its synthesis operator, then usingthis result, we obtain its stability and some important conclusions such as a q-Besselianframe is a dualizable q-frame etc. On this foundation, in section 2 we prove the perturbationtheorem of q-Besselian frame,which is a generalization of the perturbation theorem ofBesselian frame in Hilbert spaces.Chapter 4 is about q-Riesz frame and the relations among several frames. We mainlydiscuss the necessary and sufficient condition of q-Riesz frame and the stability of q-Rieszframe under perturbation. We also obtain a meaning result about perturbation. From this,we prove the perturbation theorem of Riesz frame in Hilbert spaces. At the end of this part,we discuss the relations among p-Riesz basis, dualizable q-frame, q-Besselian frame andq-Riesz frame.