多元后移不变子空间与可观测性算子
Multivariable backward-shift-invariant subspaces and observability operators
摘要：
众所周知，单位圆盘上的Hardy空间的子空间在后移下是不变的，其出现为与具有稳定状态动力学的离散时间线性系相关联的可观测性算子的图像，以及Hilbert空间收缩算子的函数模型空间。我们讨论了这种结构的两个多变量扩展，其中经典的Hardy空间被（1）形式幂级数的Fock空间在一个集合的$d$非交换不定与范数平方可和的向量系数，（2）再生核希尔伯特空间（现在通常称为Arveson空间）在单位球的${\mathbb C}^{d}$与再生核$k(λ, ζ) = 1/(1 - < λ, ζ>)$（$\lam, ζ\in {\mathbb C}^{d}$与$\| λ\|, \| ζ\| < 1$）。在第一种情况下，相关的线性系是非交换的Fornasini-Marchesini型，沿着一个自由半群与$d$生成元的发展，而在第二种情况下的线性系是一个标准的（交换）Fornasini-Marchesini型系统，沿着整数格${\mathbb Z}^{d}$的发展。阿贝尔化映射（或福克空间的对称化）将第一种情况与第二种情况联系起来。第二种情况具有特殊的功能，这取决于定义状态动态的算子元组是否是可交换的。本文主要研究多维状态输出线性系及其相应的能观性算子。
英文摘要：
It is well known that subspaces of the Hardy space over the unit disk which are invariant under the backward shift occur as the image of an observability operator associated with a discrete-time linear system with stable state-dynamics, as well as the functional-model space for a Hilbert space contraction operator. We discuss two multivariable extensions of this structure, where the classical Hardy space is replaced by (1) the Fock space of formal power series in a collection of $d$ noncommuting indeterminates with norm-square-summable vector coefficients, and (2) the reproducing kernel Hilbert space (often now called the Arveson space) over the unit ball in ${\mathbb C}^{d}$ with reproducing kernel $k(λ, ζ) = 1/(1 - < λ, ζ>)$ ($\lam, ζ\in {\mathbb C}^{d}$ with $\| λ\|, \| ζ\| < 1$). In the first case, the associated linear system is of noncommutative Fornasini-Marchesini type with evolution along a free semigroup with $d$ generators, while in the second case the linear system is a standard (commutative) Fornasini-Marchesini-type system with evolution along the integer lattice ${\mathbb Z}^{d}$. An abelianization map (or symmetrization of the Fock space) links the first case with the second. The second case has special features depending on whether the operator-tuple defining the state dynamics is commutative or not. The paper focuses on multidimensional state-output linear systems and the associated observability operators.
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