算子球的双全纯变换群的么正表示与不动点
Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls
摘要：
本文证明了从有限维希尔伯特空间到可分希尔伯特空间的算子空间的开单位球（我们称之为“算子球”），如果赋予它一个双曲度量（它是复平面上单位圆盘上标准双曲度量的类似物），则它具有一个限制形式的正规结构.利用这个结果得到了算子球的双全纯自同构群的一个不动点定理。利用不动点定理证明了可分希尔伯特空间中具有不变不定二次型且具有无穷多个负平方的有界表示是可酉化的（等价于酉表示）.我们应用这个结果来寻找Pontryagin空间中的不变子空间的对偶对。在附录中，我们提出的结果伊泰Shafrir关于双曲度量的运营商球。
英文摘要：
We show that the open unit ball of the space of operators from a finite dimensional Hilbert space into a separable Hilbert space (we call it "operator ball") has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In the appendix we present results of Itai Shafrir about hyperbolic metrics on the operator ball.
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