Arveson空间上的Schur类乘子：de Branges-Rovnyak再生核空间和交换传递函数实现
Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations
摘要：
经典Schur类的一个有趣且最近研究较多的推广是单位球${\mathbb B}^{d} \subset {\mathbb C}^{d}$上再生核希尔伯特空间${\mathcal H}(k_{d})$的压缩算子值乘子类$S$，其中$k_{d}$是${\mathbb B}^{d}$上的正核$k_{d}(λ, ζ) = 1/(1 - < λ, ζ>)$。与正核$K_{S}(λ, ζ) = (I - S(λ) S(ζ)^{*}) \cdot k_{d}(λ, ζ)$相关的再生核空间${\mathcal H}(K_{S})$是经典de Branges-Rovnyak正则模型空间的自然多变量推广。在多变量情况下出现的一个特殊特征是，空间${\mathcal H}(K_{S})$在${\mathcal H}(k_{d})$上的乘法算子$M_{λ_{j}} \colon f(λ) \mapsto λ_{j} f(λ)$的伴随$M_{λ_{j}}^{*}$下通常可能不是不变的。我们发现，${\mathcal H}(K_{S})$下$M_{λ_{j}}^{*}$为每个$j = 1, ..., d$的不变性是等价的存在一个弱共等距实现的形式$S(λ) = D + C (I - λ_{1}A_{1} ... - λ_{d} A_{d})^{-1}(λ_{1}B_{1} + ... + λ_{d} B_{d})$ $S$，使状态算子$A_{1}, ..., A_{d}$成对交换。我们表明，这种特殊的情况总是发生的情况下，内部功能$S$（其中相关的乘法算子$M_{S}$是一个部分等距），和内部乘数的特点是存在这样一个实现，使国家运营商$A_{1}, >..., A_{d}$满足一个额外的稳定性。
英文摘要：
An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers $S$ for the reproducing kernel Hilbert space ${\mathcal H}(k_{d})$ on the unit ball ${\mathbb B}^{d} \subset {\mathbb C}^{d}$, where $k_{d}$ is the positive kernel $k_{d}(λ, ζ) = 1/(1 - < λ, ζ>)$ on ${\mathbb B}^{d}$. The reproducing kernel space ${\mathcal H}(K_{S})$ associated with the positive kernel $K_{S}(λ, ζ) = (I - S(λ) S(ζ)^{*}) \cdot k_{d}(λ, ζ)$ is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space ${\mathcal H}(K_{S})$ in general may not be invariant under the adjoints $M_{λ_{j}}^{*}$ of the multiplication operators $M_{λ_{j}} \colon f(λ) \mapsto λ_{j} f(λ)$ on ${\mathcal H}(k_{d})$. We show that invariance of ${\mathcal H}(K_{S})$ under $M_{λ_{j}}^{*}$ for each $j = 1, ..., d$ is equivalent to the existence of a weakly coisometric realization for $S$ of the form $S(λ) = D + C (I - λ_{1}A_{1} ... - λ_{d} A_{d})^{-1}(λ_{1}B_{1} + ... + λ_{d} B_{d})$ such that the state operators $A_{1}, ..., A_{d}$ pairwise commute. We show that this special situation always occurs for the case of inner functions $S$ (where the associated multiplication operator $M_{S}$ is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators $A_{1}, >..., A_{d}$ satisfy an additional stability property.
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