Study of frames in Hilbert C?-modules and their representation as regular k-distance sets

Abstract

The definition of basis in the study of vector space is very antagonistic. As a resultof that, one might look for a prominent substitute. Frames are such a notion, asthe linear independence between the frame elements is not required. Further, theadditional degree of freedom coming from the structure of C?-algebra A enrichesthe theory of frames in Hilbert C?-modules. This thesis aims to introduce variousnotions of frame theory in Hilbert C?-modules as they are the subjects of the recentstudy. We also introduced the notion of a regular k-distance frame in Hilbert space.The thesis is planned to be organized into six chapters, along with the introductoryand literature survey chapter and a chapter for conclusions and the scope offuture research. Chapter 1 of the thesis is the introductory chapter, where a briefintroduction of frame theory in Hilbert space as well as in Hilbert C?-module hasbeen discussed. The interest in taking the particular research problem has beenoutlined. A concise but sufficient literature survey has been presented. In Chapter2, we introduced the concept of a regular k-distance frame in Hilbert space as wellas focused on k-distance tight frames for the underlying space. We have introducedthe definition of dual frames for a regular k-distance set. Finally, the perturbationresult for regular k-distance frames is established. The objective of Chapter 3 is tointroduce woven g-frames in Hilbert C?-modules and to develop its fundamentalproperties. This study establishes sufficient conditions under which two g-framespossess weaving properties. We also investigated the sufficient conditions underwhich a family of g-frames includes weaving properties. Chapter 4 is concernedwith weaving K-frames in Hilbert C?-module. We introduced the concept ofweaving K-frames and defined an atomic system for weaving K-frames in HilbertC?-module. We studied weaving K-frames in this chapter from the operator theoretic approach. Moreover, we gave an equivalent definition for weaving Kframes.In Chapter 5, we introduced the notion of a controlled K-frame in HilbertC?-modules. We established the equivalent condition for controlled K-frame inHilbert C?-modules. We investigated some operator theoretic characterizations ofcontrolled K-frames and controlled Bessel sequences. Moreover, we established therelationship between the K-frames and controlled K-frames. We also investigatedthe invariance of a C-controlled K-frame under a suitable map T. At last, weproved a perturbation result for controlled K-frame in Hilbert C?-modules.An equivalent definition is much easier to apply and permits us to study the varioustypes of frames from the operator theory point of view. The multiple notions offrame theory developed in this thesis will draw the attention of researchers to workin this area. At last, in Chapter 6, we summarize all the work that has been doneso far and feature the potential avenues for the future scope of research.Motivation and Objective of the ThesisIn a vector space, a set of vectors is referred to as a basis if every element in theunderlying space can be expressed in terms of a finite linear combination of thebasis vectors uniquely. The definition of basis in the study of vector space is veryantagonistic. As a result of that, one might look for a prominent substitute. Framesare such a notion as the linear independence between the frame elements is notrequired. In addition to that, the additional degree of freedom coming from thestructure of C?-algebra A enriches the theory of frames in Hilbert C?-modules.We intend to see whether the results of frame theory in Hilbert spaces hold forframe theory in Hilbert C?-modules and, if not, then to study what modificationswe need. In Chapter 2, we investigated the concept of a regular k-distance framein Hilbert space which is the extension of a regular two-distance frame in Hilbertspace. A regular two-distance frame is a particular type category of frame whichhas some nice properties. Motivated by this, we studied regular k-distance frames,in particular, regular tight k-distance frames in Hilbert space. Tight frames are thosein which the lower and upper frame bounds are equal. Tight frames play a key rolein wide applications as tight frames look like a more natural way to reconstruct vectors. Tight frames are closest to orthonormal bases as they are a redundant setof vectors and have properties like basis. In Chapter 3 and Chapter 4, we studiedthe concept of woven frames in Hilbert C?-modules. Recently many people gotsignificant results in frame theory by generalizing the results which are present inHilbert space to Hilbert C?-modules. The concept of weaving frames is applicablein wireless sensor networks that require distributed processing under differentframes, as well as pre-processing of signals using Gabor frames. Generalizedframes (or g-frames) include standard frames, bounded invertible linear operators,and many recent generalizations of frames. g-frames in Hilbert C?-modules interestmany useful properties with their comparable tools in Hilbert space. As we know,K-frames and standard frames diverge in many aspects; we introduce the conceptof weaving K-frames and define an atomic system for weaving K-frames in HilbertC?-modules. As it is easier to work, we gave an equivalent definition for weavingK-frames and characterized weaving K-frames from the operator theory point ofview. In Chapter 5, we introduced the notion of controlled K-frames in Hilbert C?-modules. Controlled frames have been an area of interest because of their expertisein improving the numerical efficiency of iterative algorithms for inverting theframe operator.