In this work we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space

中文翻译：

---

全纯和切片超全纯函数的实现：Kerin 空间案例

在这项工作中，我们处理算子值函数的实现结果，这些函数在复数意义上是解析的或四元数上的切片超全纯。在复杂的设置中，我们证明了具有等距状态空间算子的原点邻域中算子值函数解析的实现定理，从而推广了幺正情况下的类似结果。与以前作品的主要区别是使用了再现内核 Kerin 空间。然后我们在四元数设置中证明这个结果的对应物。目前的工作是第一篇论文，它提出了一个具有四元数 Kerin 空间的状态空间的实现定理