多维线性耗散系统的W ~*-对应方法
A W*-correspondence approach to multi-dimensional linear dissipative systems
摘要：
算子代数学家P. Muhly和B. Solel，主要是由算子代数和数学物理理论的动机，描绘了一个一般的抽象框架，其中系统理论的想法出现在伪装的形式。这些系统理论成分包括：输入/状态/输出线性系的系统矩阵，从“时域“到“频域“的Z变换，以及由应用于初始条件的观测函数加上应用于输入信号的Z变换的传递函数给出的输出信号的Z变换。在这里，我们列出了一般Muhly-Solel形式主义的定义和主要结果，并针对两种特定类型的多维线性系来说明它们：（1）具有传递函数等于${\mathbb C}^d$中单位球上的全纯算子值函数的耗散Fornasini-Marchesini状态空间表示，（2）Hilbert空间上沿自由半群发展且传递函数定义在严格行压缩的非对易球上的非对易耗散Fornasini-Marchsini线性系。
英文摘要：
Recent work of the operator algebraists P. Muhly and B. Solel, primarily motivated by the theory of operator algebras and mathematical physics, delineates a general abstract framework where system theory ideas appear in disguised form. These system-theory ingredients include: system matrix for an input/state/output linear system, Z-transform from a "time domain" to a "frequency domain", and Z-transform of the output signal given by an observation function applied to the initial condition plus a transfer function applied to the Z-transform of the input signal. Here we set down the definitions and main results for the general Muhly-Solel formalism and illustrate them for two specific types of multi-dimensional linear systems: (1) dissipative Fornasini-Marchesini state-space representations with transfer function equal to a holomorphic operator-valued function on the unit ball in ${\mathbb C}^d$, and (2) noncommutative dissipative Fornasini-Marchsini linear systems with evolution along a free semigroup and with transfer function defined on the noncommutative ball of strict row contractions on a Hilbert space.
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