Let Y be a complex Banach space and let \(r\ge 1\). In this paper, we are concerned with an extension operator \(\varPhi _{\alpha , \beta }\) that provides a way of extending a locally univalent function f on the unit disc \(\mathbb {U}\) to a locally biholomorphic mapping \(F\in H(\varOmega _r)\), where \(\varOmega _r=\{(z_1,w)\in \mathbb {C}\times Y: |z_1|^2+\Vert w\Vert _Y^r<1\}\). We prove that if f can be embedded as the first element of a g-Loewner chain on \(\mathbb {U}\), where g is a convex (univalent) function on \(\mathbb {U}\) such that \(g(0)=1\) and \(\mathfrak {R}g(\zeta )>0\), \(\zeta \in \mathbb {U}\), then \(F =\varPhi _{\alpha , \beta }(f)\) can be embedded as the first element of a g-Loewner chain on \(\varOmega _r\), for \(\alpha \in [0, 1]\), \(\beta \in [0, 1/r]\), \(\alpha +\beta \le 1\). We also show that normalized univalent Bloch functions on \(\mathbb {U}\) (resp. normalized uniformly locally univalent Bloch functions on \(\mathbb {U}\)) are extended to Bloch mappings on \(\varOmega _r\) by \(\varPhi _{\alpha ,\beta }\), for \(\alpha >0\) and \(\beta \in [0,1/r)\) (resp. for \(\alpha =0\) and \(\beta \in [0,1/r]\)). In the case of the Muir type extension operator \(\varPhi _{P_k}\), where \(k\ge 2\) is an integer and \(P_k:Y\rightarrow \mathbb {C}\) is a homogeneous polynomial mapping of degree k with \(\Vert P_k\Vert \le d(1,\partial g(\mathbb {U}))/4\), we prove a similar extension result for the first elements of g-Loewner chains on \(\varOmega _k\). Next, we consider a modification of the Muir type extension operator \(\varPhi _{G,k}\), where \(k\ge 2\) is an integer and \(G:Y\rightarrow \mathbb {C}\) is a holomorphic function such that \(G(0)=0\) and \(DG(0)=0\), and prove that if g is a univalent function with real coefficients on \(\mathbb {U}\) such that \(g(0)=1\), \(\mathfrak {R}g(\zeta )>0\), \(\zeta \in \mathbb {U}\), and g satisfies a natural boundary condition, and if the extension operator \(\varPhi _{G,k}\) maps g-starlike functions from the unit disc \(\mathbb {U}\) into starlike mappings on \(\varOmega _k\), then G must be a homogeneous polynomial of degree at most k. We also obtain a preservation result of normalized uniformly locally univalent Bloch functions on \(\mathbb {U}\) to Bloch mappings on \(\varOmega _k\) by \(\varPhi _{P_k}\).

中文翻译：

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复杂Banach空间中的g -Loewner链，Bloch函数和扩展算子

令Y为复Banach空间，令\（r \ ge 1 \）。在本文中，我们关注扩展运算符\（\ varPhi _ {\ alpha，\ beta} \），它提供了一种将单位圆盘\（\ mathbb {U} \）上的局部单价函数f扩展为局部双全纯映射\（F \ in H（\ varOmega _r）\），其中\（\ varOmega _r = \ {（z_1，w）\ in \ mathbb {C} \ times Y：| z_1 | ^ 2 + \垂直w \ Vert _Y ^ r <1 \} \）。我们证明了如果˚F可以嵌入作为第一个元素克上-Loewner链\（\ mathbb【U} \） ，其中克是上凸的（一价）函数\（\ mathbb【U} \）使得\（克（0）= 1 \）和\（\ mathfrak {R}克（\ζ电）> 0 \） ，\（\ζ电\在\ mathbb【U} \） ，然后\（F = \可以将varPhi _ {\\ alpha，\ beta}（f）\）嵌入为\（\ varOmega _r \）上g -Loewner链的第一个元素，对于\（\ alpha \ in [0，1] \），\（\ beta \ in [0，1 / r] \），\（\ alpha + \ beta \ le 1 \）。我们还表明在该归一化的一价布洛赫功能\（\ mathbb【U} \） （相应的归一化一致局部单价布洛赫功能\（\ mathbb【U} \） ）延伸到布洛赫映射上\（\ varOmega _r \ ）由\（\ varPhi _ {\ alpha，\ beta} \），对于\（\ alpha> 0 \）和\（\ beta \ in [0,1 / r）\）（分别针对\（\ alpha = 0 \）和\（\ beta \ in [0,1 / r] \））。对于Muir类型扩展运算符\（\ varPhi _ {P_k} \），其中\（k \ ge 2 \）是整数，\（P_k：Y \ rightarrow \ mathbb {C} \）是齐次的用\（\ Vert P_k \ Vert \ le d（1，\ partial g（\ mathbb {U}））/ 4 \）对度k进行多项式映射，我们证明了g -Loewner链的第一个元素的相似扩展结果在\（\ varOmega _k \）上。接下来，我们考虑对Muir类型扩展运算符\（\ varPhi _ {G，k} \）的修改，其中\（k \ ge 2 \）是整数，\（G：Y \ rightarrow \ mathbb {C} \）是全纯函数，使得\（G（0）= 0 \）和\（DG（0 ）= 0 \），并证明g是\（\ mathbb {U} \）上具有实系数的单价函数，使得\（g（0）= 1 \），\（\ mathfrak {R} g（ \ζ电）> 0 \） ，\（\ζ电\在\ mathbb【U} \） ，和克满足的自然边界条件，并且如果扩展操作者\（\ varPhi _ {G，K} \）映射克-从单位光盘\（\ mathbb {U} \）的星形函数转换为\（\ varOmega _k \）上的星形映射，则G必须是至多k的齐次多项式。我们还通过\（\ varPhi _ {P_k} \）获得\（\ mathbb {U} \）上归一化的统一局部单价Bloch函数到\（\ varOmega _k \）上Bloch映射的标准化保存结果。