论文标题
通勤希尔伯特空间收缩的功能模型
Functional Models for Commuting Hilbert-space Contractions
论文作者
论文摘要
我们开发了一个sz.-nagy-foias-type功能模型，用于交换承包操作员元组$ \下划线{t} =（t_1，\ dots，t_d）$，具有$ t = t_1 \ cdots t_d $等于完全非独立的收缩。我们确定其他不变的$ {\ Mathbb g} _ \ sharp，{\ Mathbb w} _ \ sharp $，除了Sz.-nagy--foias--foias特征函数$θ_t$ for产品操作员$ t $ $θ_t$成为原始操作员元组$ \ usepline {t} $的完全统一不变。对于总体上，$ d \ ge 3 $总体上没有$ \ usepline {t} $的换向等距提升； however there is a (not necessarily commutative) isometric lift having some additional structure so that, when compressed to the minimal isometric-lift space for the product operator $T$, generates a special kind of lift of $\underline{T}$, herein called a {\em pseudo-commutative contractive lift} of $\underline{T}$, which in turn leads to the functional model for $ \ usewinline {t} $。这项工作与最近开发的模型理论具有许多相似之处，该理论具有对对称性的bidisk收缩（可交换操作员对$（s，p）$，具有对称的bidisk $γ$作为频谱集）和四载体收缩（通勤操作员$（a，b，p）$具有tetrablock domain domainabock domainain $ {b，b，p）$ {\ nmath $ {\ nmath $ {\ a.
We develop a Sz.-Nagy--Foias-type functional model for a commutative contractive operator tuple $\underline{T} = (T_1, \dots, T_d)$ having $T = T_1 \cdots T_d$ equal to a completely nonunitary contraction. We identify additional invariants ${\mathbb G}_\sharp, {\mathbb W}_\sharp$ in addition to the Sz.-Nagy--Foias characteristic function $Θ_T$ for the product operator $T$ so that the combined triple $({\mathbb G}_\sharp, {\mathbb W}_\sharp, Θ_T)$ becomes a complete unitary invariant for the original operator tuple $\underline{T}$. For the case $d \ge 3$ in general there is no commutative isometric lift of $\underline{T}$; however there is a (not necessarily commutative) isometric lift having some additional structure so that, when compressed to the minimal isometric-lift space for the product operator $T$, generates a special kind of lift of $\underline{T}$, herein called a {\em pseudo-commutative contractive lift} of $\underline{T}$, which in turn leads to the functional model for $\underline{T}$. This work has many parallels with recently developed model theories for symmetrized-bidisk contractions (commutative operator pairs $(S,P)$ having the symmetrized bidisk $Γ$ as a spectral set) and for tetrablock contractions (commutative operator triples $(A, B, P)$ having the tetrablock domain ${\mathbb E}$ as a spectral set).
