Collections of operators or matrices

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Science is the engine that drives technology. Breakthroughs in basic science often lead to new technologies. Mathematics is the framework that supports most science. Without the structured form imposed by mathematics, it is extremely difficult to understand complicated phenomena. Applied mathematics is the study of mathematical questions that arise from consideration of some scientific phenomena. Science advances so fast that the time lag between posing the mathematical questions that arise from science until the time when sophisticated mathematical solutions can be applied to science is often the major factor slowing the advancement of knowledge. One solution to this dilemma is to have mathematics ready when the need arises. This is one justification of the need to research and fund pure mathematics. Pure mathematics is the study of mathematical structures with no specific application in mind. There are many examples of pure mathematical theories which were developed because of their beauty or intrinsic interest and that have proved to be of major use to a developing scientific field. One such example is the application of number theory to public key cryptography, which now forms the basis for secure communication on the Internet. In my research, I study collections of operators and matrices. These have been used to model phenomena in quantum mechanics, control theory, signal processing and numerous other areas. The exciting new area of Quantum Computing relies heavily on knowledge of the structure of operators and matrices. Basic mathematical information about the structure of bounded operators may lead to advances in these fields, and others yet to be even studied. The first step in analysis of structure is to understand the basic objects from which the structure is built and determine how the structure is built from basic objects. We consider such problems for collections of operators satisfying certain properties.