Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc
Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc
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对称 Bidisc 中的测地线、缩回和保范扩展特性
DOI：
10.1090/memo/1242
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Agler J
中科院分区：
文献类型：
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作者：
Agler J
定义在上的集合具有保模扩张性质，如果上的每一个有界全纯函数都有一个上确界范数相同的全纯扩张。我们证明了一个代数子集的对称bidischas保范扩张性质当且仅当它是一个单例，本身，复测地线，或工会的集合和复测地线度。我们还证明了复测地线与中的非平凡全纯收缩不重合。因此，与球或双圆盘的情况相反，存在具有保范扩张性质但不是的全纯收缩的集合。在证明过程中，我们得到了模自同构中复测地线的详细分类。我们给冯诺依曼型不等式的应用-压缩（即，交换对运营商的封闭是一个谱集）和对称函数的交换对压缩运营商。我们发现其他三个域，包含集的范数保持扩展属性，这是不收回：他们是谱球矩阵，tetrablock和pentablock。我们还确定了子集的bidisc具有保范扩展属性的对称函数。
A setin a domaininhas the norm-preserving extension property if every bounded holomorphic function onhas a holomorphic extension towith the same supremum norm. We prove that an algebraic subset of the symmetrized bidischas the norm-preserving extension property if and only if it is either a singleton,itself, a complex geodesic of, or the union of the setand a complex geodesic of degreein. We also prove that the complex geodesics incoincide with the nontrivial holomorphic retracts in. Thus, in contrast to the case of the ball or the bidisc, there are sets inwhich have the norm-preserving extension property but are not holomorphic retracts of. In the course of the proof we obtain a detailed classification of the complex geodesics inmodulo automorphisms of. We give applications to von Neumann-type inequalities for-contractions (that is, commuting pairs of operators for which the closure ofis a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball ofmatrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions.