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Loewner chains, Bloch mappings and Pfaltzgraff-Suffridge extension operators on bounded symmetric domains. (English) Zbl 1436.32080

In the paper, positive answers to the following two questions are given:
“Question 1.1: Let \(\mathbb{B}_X\) be a bounded symmetric domain in \(X=\mathbb{C}^n\) and let \(Y\) be an arbitrary complex Banach space. Are there a domain \(\mathbb{D}\subset Z=X\times Y\) such that \(\mathbb{B}_X\times\left\{0\right\}\subset\mathbb{D}\), and an extension operator \(\Psi:\mathcal{L}S(\mathbb{B}_X)\rightarrow\mathcal{L}S(\mathbb{D})\) such that \(\Psi\) extends the first elements of Loewner chains from the unit ball \(\mathbb{B}_X\) to the first elements of Loewner chains on the domain \(\mathbb{D}\)?”
“Question 1.2: Let \(\mathbb{B}_X\) be a bounded symmetric domain in \(X=\mathbb{C}^n\) and let \(Y\) be an arbitrary complex Banach space. Are there a domain \(\mathbb{D}\subset Z=X\times Y\) such that \(\mathbb{B}_X\times\left\{0\right\}\subset\mathbb{D}\), and an extension operator \(\Psi:\mathcal{L}S(\mathbb{B}_X)\rightarrow\mathcal{L}S(\mathbb{D})\) such that \(\Psi\) extends normalized locally univalent Bloch mappings on \(\mathbb{B}_X\) to locally univalent Bloch mappings on the domain \(\mathbb{D}\)?”

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
32A18 Bloch functions, normal functions of several complex variables
32A30 Other generalizations of function theory of one complex variable
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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