The vector-valued Stieltjes moment problem with general exponents

We characterize the sequences of complex numbers $(z_n{)}_{n\in \mathbb{N}}$ and the locally complete $(DF)$-spaces $E$ such that for each $(e_n{)}_{n\in \mathbb{N}}\in E^{\mathbb{N}}$ there exists an $E$-valued function $\mathbf{f}$ on $(0,\mathrm{\infty })$ (satisfying a mild regularity condition) such that $${\int }_0^{\mathrm{\infty }}{t}^{z_n}\mathbf{f}(t)dt=e_n,\mathrm{\forall }n\in \mathbb{N},$$ where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution $\mathbf{f}$ that is smooth on $(0,\mathrm{\infty })$ and satisfies certain optimal growth bounds near $0$ and $\mathrm{\infty }$. The scalar-valued case $(E=\mathbb{C})$ was treated by Durán [Math. Nachr. 158 (1992), 175-194]. Our work is based upon his result.