Topics in Dilation Theory

PI：Scott McCullough提案编号：0140112摘要Nagy伸缩定理将Hilbert空间上的压缩算子描述为限制在不变子空间上的等距算子的伴随，对算子理论及其应用产生了深远的影响。 表示族和边界表示族（不一定是不可约的）的Agler边界的存在性和算子代数的C-星包络是Sz.-的抽象推广。Nagy膨胀定理，自然编码Pick插值和在许多情况下交换提升。它们也为Beurling定理的版本提供了一个自然的设置。 Agler边界和C-星包络已明确计算的家庭的完全压缩表示的乘子代数的对称Fock空间，代数的Dirichlet空间，以及concernents的盘和环代数。PI将调查的情况下，强版本的挑选插值，Beurling定理，和交换liftinerados作为一个功能的复杂性相应的Agler边界和C-星信封。调查将与函数理论，特殊功能，微分方程和一般理论的算子代数。Thre也计划继续学习因式分解的多项式在几个非交换变量追求的主题，一个变量的结果最自然地推广到非交换setting.Operator理论有丰富的相互作用的历史与工程和物理以及其他重要领域的数学，包括复变函数理论和代数几何。算子理论、算子代数和算子系统最初是作为研究物理学中出现的积分和微分方程的工具而开发的，在现代量子物理学中发挥着重要作用。算子理论的四个基本主题，Sz。纳吉膨胀、皮克插值、交换子提升和伯林定理现在是系统理论中的基本技术，而这些技术又在图像处理和控制理论中有重要的应用-自动控制器（如自动驾驶仪）背后的数学。在拟议的工作中的一个主要重点是概括这些主题，以期对系统理论，控制理论，算子代数和函数理论的应用。麦卡洛还将研究因式分解问题的多项式在几个非交换变量。(Here xy可能不等于yx。）这项研究预计将与非交换代数几何和线性矩阵不等式或LMI的，经常出现在工程应用中的连接。

PI: Scott McCulloughProposal Number: 0140112ABSTRACTThe Sz. Nagy Dilation Theorem, which models a contractionoperator on Hilbert space as the adjoint of an isometry restricted to an invariant subspace has had a profound influence on operator theory and its applications. The existence of the Agler boundary for families of representations and of boundary representations (not necessarily irreducible) and the C-star envelope for operator algebras are abstract generalizations of the Sz.-Nagy dilation theorem which naturally encode Pick interpolation and in many cases commutant lifting. They also provide a natural setting for versions of Beurling's theorem. The Agler boundary and C-star envelope have been explicitly computed for the families of completely contractive representations of the algebra of multipliers of symmetric Fock space, the algebra of multipliers of the Dirichlet space, as well as quotients of the disc and annulus algebra. The PI will investigate situations in which strong versions of Pick interpolation, Beurling's theorem, and commutant liftinghold as a function of the complexity of the correspondingAgler boundary and C-star envelope. The investigation will make contact with function theory, special functions, differential equations, and the general theory of operator algebras. Thre are also plans to continue studyingfactorization of polynomials in several non-commuting variables pursuing the theme that one-variable resultsmost naturally generalize to the non-commutative setting.Operator theory has a history of rich interplay with engineering and physics as well as other vital areas of mathematics including complex function theory and algebraic geometry. Originally developed as a tool to study integral and differential equations arising in physics, operator theory, operator algebras, and operator systems play an important role in modern quantum physics. Four fundamental themes in operator theory, Sz.-Nagy dilation, Pick interpolation, commutant lifting, and Beurling's theorem are now basic technology in systems theory which in turn has important applications in image processing and control theory - the mathematics behind automatic controllers such as autopilots. A major emphasis in the proposed work is generalizations of these themes with a view toward applications to systems theory, control theory, operator algebras, and function theory. McCullough will also study factorization problems for polynomials in several non-commuting variables. (Here xy may not equal yx.) This investigation isexpected to make connection with non-commutative algebraic geometry and Linear Matrix Inequalities or LMI's which frequently arise in engineering applications.