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 $Df(0,t)=e^{tA}$, where $A\in L(X,X)$ is such that $k_+(A)<2m(A)$, on the unit ball of a separable reflexive complex Banach space $X$. We also give improvements of the results obtained recently by Hamada and Kohr, but we omit their proofs for the sake of brevity. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings $v(z,s,t)$ with normalization $Dv(0,s,t)=e^{-(t-s)A}$ for $t\geq s\geq 0$, where $m(A)>0$, 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 $Df(0,t)=e^{tA}$ 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.