In this paper we focus our attention on the Graham–Kohr extension operator \(\Psi _{\alpha }\) (introduced by Graham and Kohr in Complex Var. Theory Appl. 47:59–72, 2002) defined by \(\begin{aligned} \Psi _{\alpha }(f)(z_1,w)=\bigg (f(z_1),\bigg (\dfrac{f(z_1)}{z_1}\bigg )^{\alpha }w \bigg ), \quad (z_1,w)\in \Omega _{p,r}, \, \alpha \in [0,1] \end{aligned}\) and its properties on the domain \(\Omega _{p,r}=\{(z_1,w)\in \mathbb {C}\times X:\vert z_1\vert ^p+\Vert w\Vert ^r_X<1\}\) , where X is a complex Banach space and \(p,r\ge {1}\) . In the main part of the paper we will prove that the Graham–Kohr extension operator \(\Psi _{\alpha }\) preserves the first elements of g-Loewner chains from the unit disc U to the domain \(\Omega _{p,r}\) , where \(p,r\ge {1}\) and \(\alpha \in [0,1]\) . As consequences of this result, we obtain that the Graham–Kohr extension operator preserves, in particular, the first elements of Loewner chains, the g-starlike mappings, the strongly starlike mappings of order \(d\in (0,1]\) , the parabolic starlike mappings of order \(d\in [0,1)\) , the starlike mappings of complex order \(\lambda \) , the almost starlike mappings of order \(\mu \in [0,1)\) , respectively the spirallike mappings of order \(\delta \in (-\pi /2,\pi /2)\) on the same domain \(\Omega _{p,r}\) for \(p,r\ge {1}\) .
