Topics in Dilation Theory

PI: Scott mccullough提案编号：0140112abstract Sz。Nagy膨胀定理将Hilbert空间上的收缩算子建模为限定在不变子空间上的等距的伴随算子，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（不一定不可约）的Agler边界的存在性以及算子代数的c星包络是Sz的抽象推广。-纳吉膨胀定理，自然编码匹克插值和在许多情况下的交换提升。它们也为伯灵定理的版本提供了一个自然的背景。对于对称Fock空间的乘子代数、Dirichlet空间的乘子代数以及盘和环代数的商的完全压缩表示族，明确地计算了Agler边界和c星包络。PI将研究强版本的Pick插值、Beurling定理和交换子提升作为相应agler边界和c星包络复杂性的函数的情况。研究将涉及函数理论、特殊函数、微分方程和算子代数的一般理论。也有计划继续研究多项式在几个非交换变量中的因式分解，追求单变量结果最自然地推广到非交换设置的主题。算符理论与工程和物理以及其他重要的数学领域，包括复函数理论和代数几何，有着丰富的相互作用的历史。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方程和微分方程的工具而发展起来的，在现代量子物理学中起着重要的作用。算符理论的四个基本主题。nagy膨胀、Pick插值、换向提升和Beurling定理现在是系统理论的基本技术，它们反过来在图像处理和控制理论（自动控制器背后的数学，如自动驾驶仪）中有重要的应用。在提议的工作中，一个主要的重点是将这些主题概括为系统理论、控制理论、算子代数和函数理论的应用。McCullough还将研究几个非交换变量多项式的因式分解问题。（这里xy可能不等于yx。）本研究旨在与非交换代数几何和线性矩阵不等式（LMI）联系起来，这些问题在工程应用中经常出现。

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.