对称双圆盘中的测地线、收缩和保范延拓性质
Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc
摘要：
在$\mathbb{C}^n$中的定义域$U$中的集合$V$具有{ \em保模扩张性质}，如果$V$上的每个有界全纯函数都具有$U$的具有相同上确界范数的全纯扩张。我们证明了{ \em对称双圆盘} \[ G := \{(z+w,zw):|z|<1, |w| < 1 \} \]的代数子集具有保范扩张性质当且仅当它是一个单点，$G$本身，$G$的复测地线，或集合$\{(2z,z^2): |z|<1\}$和$G$中次数$1$的复测地线的并集.我们还证明了$G$中的复测地线与$G$中的非平凡全纯收缩相一致。因此，与球或双圆盘的情况相反，在$G$中存在具有保范扩张性质但不是$G$的全纯收缩的集合。在证明过程中，我们得到了$G$的$G$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$Γ$ -压缩（即交换对运营商的封闭的$G$是一个谱集）和对称函数的交换对压缩运营商的应用。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：他们是谱球的$2\times 2$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。
英文摘要：
A set $V$ in a domain $U$ in $\mathbb{C}^n$ has the {\em norm-preserving extension property} if every bounded holomorphic function on $V$ has a holomorphic extension to $U$ with the same supremum norm. We prove that an algebraic subset of the {\em symmetrized bidisc} \[ G := \{(z+w,zw):|z|<1, |w| < 1 \} \] has the norm-preserving extension property if and only if it is either a singleton, $G$ itself, a complex geodesic of $G$, or the union of the set $\{(2z,z^2): |z|<1\}$ and a complex geodesic of degree $1$ in $G$. We also prove that the complex geodesics in $G$ coincide with the nontrivial holomorphic retracts in $G$. Thus, in contrast to the case of the ball or the bidisc, there are sets in $G$ which have the norm-preserving extension property but are not holomorphic retracts of $G$. In the course of the proof we obtain a detailed classification of the complex geodesics in $G$ modulo automorphisms of $G$. We give applications to von Neumann-type inequalities for $Γ$-contractions (that is, commuting pairs of operators for which the closure of $G$ is 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 of $2\times 2$ matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions.
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