论文标题
四足动收缩的扩张理论和功能模型
Dilation theory and functional models for tetrablock contractions
论文作者
论文摘要
Sz.-Nagy的经典结果断言，可以将Hilbert Space Cartraction运算符$ T $扩张到单一$ \ cu $。这些想法的更通用的多变量设置是设置，即（i）单位磁盘被$ {\ Mathbb c}^d $中包含的域$ω$替换，（ii）收缩操作员$ t $被通勤的元环$ \ bft =（t_1，t_1，dots，t_d） r（t_1，\ dots，t_d）\ | _ {\ cl（\ ch）} \ le \ sup _ {\ lam \ inω} | r(\lam) |$ for all rational functions with no singularities in $\overlineΩ$ and the unitary operator $\cU$ is replaced by an $Ω$-unitary operator tuple, i.e., a commutative operator $d$-tuple $\bfU = (U_1, \dots, U_d)$ of commuting normal operators with joint spectrum contained in the distinguished boundary $bΩ$的$ω$。对于给定的域$ω\ subset {\ Mathbb c}^d $，{\ em合理的扩张问题}问：给定$ \ ch $上的$ω$ - contraction $ \ bft $，是否总是有可能在任何$ω$ \ bfu $ chilbert $ ch $ c.ck \ ck \ ck \ ck ck \ ck \ ck ck \ ck \ c.ck ck ck \ ck ck ck \ ck ck ck ck \ ck ck ck ck \ c. $ d $ -varia-varia-variable函数在$ \overlineΩ$中没有奇点，可以恢复$ r（t）$ as $ r（t）= p_ \ ch r（\ bfu）| _ \ ch $。我们在这里专注于$ω$是{\ em tetrableock}的情况。 （i）我们为$ {\ mathbb e} $ - 收缩$（a，b，t）$确定一组完整的单一不变性，然后可以用来写下$（a，b，t）$的功能模型，从而扩展了早期的结果，从而扩展了为特殊情况所做的，（ii）我们确定了{\ em pseo $ pseud $} $ commuties $ {\ em-psey $ f。 （先验比$ {\ Mathbb e} $ - 等法略大），可以提高任何$ {\ mathbb e} $ - 收缩的任何$ {\ Mathbb e} $ - （iii）我们使用函数模型来恢复$ {$ {\ MathBB e} $ - iSomet-ismet-ismet fipt $（v_1）的早期结果和唯一性的结果。 $ {\ mathbb e} $ - 收缩$（a，b，t）$。
A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary $\cU$. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain $Ω$ contained in ${\mathbb C}^d$, (ii) the contraction operator $T$ is replaced by a commuting tuple $\bfT = (T_1, \dots, T_d)$ such that $\| r(T_1, \dots, T_d) \|_{\cL(\cH)} \le \sup_{\lam \in Ω} | r(\lam) |$ for all rational functions with no singularities in $\overlineΩ$ and the unitary operator $\cU$ is replaced by an $Ω$-unitary operator tuple, i.e., a commutative operator $d$-tuple $\bfU = (U_1, \dots, U_d)$ of commuting normal operators with joint spectrum contained in the distinguished boundary $bΩ$ of $Ω$. For a given domain $Ω\subset {\mathbb C}^d$, the {\em rational dilation question} asks: given an $Ω$-contraction $\bfT$ on $\cH$, is it always possible to find an $Ω$-unitary $\bfU$ on a larger Hilbert space $\cK \supset \cH$ so that, for any $d$-variable rational function without singularities in $\overlineΩ$, one can recover $r(T)$ as $r(T) = P_\cH r(\bfU)|_\cH$. We focus here on the case where $Ω$ is the {\em tetrablock}. (i) We identify a complete set of unitary invariants for a ${\mathbb E}$-contraction $(A,B,T)$ which can then be used to write down a functional model for $(A,B,T)$, thereby extending earlier results only done for a special case, (ii) we identify the class of {\em pseudo-commutative ${\mathbb E}$-isometries} (a priori slightly larger than the class of ${\mathbb E}$-isometries) to which any ${\mathbb E}$-contraction can be lifted, and (iii) we use our functional model to recover an earlier result on the existence and uniqueness of a ${\mathbb E}$-isometric lift $(V_1, V_2, V_3)$ of a special type for a ${\mathbb E}$-contraction $(A,B,T)$.
