Collections of operators or matrices

科学是驱动技术的引擎。基础科学的突破往往会带来新技术。数学是支撑大多数科学的框架。没有数学强加的结构化形式，要理解复杂的现象是极其困难的。应用数学是研究一些科学现象所产生的数学问题的学科。科学发展如此之快，以至于从提出科学产生的数学问题到复杂的数学解决方案可以应用于科学的时间间隔往往是减缓知识进步的主要因素。解决这一困境的一种方法是在需要时准备好数学。这是研究和资助纯数学的必要性的一个理由。 纯数学是对数学结构的研究，在头脑中没有具体的应用。有许多纯数学理论的例子，它们是因为它们的美丽或内在的兴趣而发展起来的，并已被证明对一个发展中的科学领域有重大用途。一个这样的例子是数论在公钥密码学中的应用，公钥密码学现在构成了互联网上安全通信的基础。 在我的研究中，我研究了运算符和矩阵的集合。它们已被用于对量子力学、控制论、信号处理和许多其他领域的现象进行建模。量子计算这一令人兴奋的新领域在很大程度上依赖于对运算符和矩阵结构的了解。关于有界算子结构的基本数学信息可能会导致这些领域的进步，甚至还有其他尚未研究的领域。 分析结构的第一步是了解用来构建结构的基本对象，并确定如何从基本对象构建结构。我们考虑满足某些性质的算子集合的这类问题。

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.