Loewner PDE in Infinite Dimensions

In this paper, we prove the existence and uniqueness of the solution f(z, t) of the Loewner PDE with normalization \(Df(0,t)=e^{tA}\) $Df(0,t)=e^{\mathit{tA}}$ , where \(A\in L(X,X)\) $A\in L(X,X)$ is such that \(k_+(A)<2m(A)\) $k_{+}(A)<2m(A)$ , on the unit ball of a separable reflexive complex Banach space X. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings v(z, s, t) with normalization \(Dv(0,s,t)=e^{-(t-s)A}\) $Dv(0,s,t)=e^{-(t-s)A}$ for \(t\ge s\ge 0\) $t\ge s\ge 0$ , where \(m(A)>0\) $m(A)>0$ , which satisfy the semigroup property on the unit ball of a complex Banach space X. We further obtain the biholomorphicity of A-normalized univalent subordination chains under some normality condition on the unit ball of a reflexive complex Banach space X. We prove the existence of the biholomorphic solutions f(z, t) of the Loewner PDE with normalization \(Df(0,t)=e^{tA}\) $Df(0,t)=e^{\mathit{tA}}$ on the unit ball of a separable reflexive complex Banach space X. The results obtained in this paper give some positive answers to the open problems and conjectures proposed by the authors in 2013.