Around Widder's characterization of the Laplace transform of an element of ${\displaystyle L^{\infty }\left({ℝ}^{+}\right)}$

Let ϰ be a positive, continuous, submultiplicative function on ${\displaystyle {ℝ}^{+}}$ such that ${\displaystyle \underset{t\to \infty }{lim}{e}^{-\omega t}{t}^{-\alpha }\varkappa \left(t\right)=a}$ for some ω ∈ ℝ, α ∈ ${\displaystyle \overline{{ℝ}^{+}}}$ and ${\displaystyle a\in {ℝ}^{+}}$. For every λ ∈ (ω,∞) let ${\displaystyle {\varphi }_{\lambda }\left(t\right)=e^{-\lambda t}}$ for ${\displaystyle t\in {ℝ}^{+}}$. Let ${\displaystyle L^1\_\{\varkappa \}\left({ℝ}^{+}\right)}$ be the space of functions Lebesgue integrable on ${\displaystyle {ℝ}^{+}}$ with weight ${\displaystyle \varkappa }$, and let E be a Banach space. Consider the map ${\displaystyle {\varphi }_{•}:(\omega ,\infty )\ni \lambda \to {\varphi }_{\lambda }\in L_{\varkappa }^1\left({ℝ}^{+}\right)}$. Theorem 5.1 of the present paper characterizes the range of the linear map ${\displaystyle T\to T{\varphi }_{•}}$ defined on ${\displaystyle L(L_{\varkappa }^1\left({ℝ}^{+}\right);E)}$, generalizing a result established by B. Hennig and F. Neubrander for ${\displaystyle \varkappa \left(t\right)=e^{\omega t}}$. If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder's characterization of the Laplace transform of a function in ${\displaystyle L^{\infty }\left({ℝ}^{+}\right)}$. Some applications of Theorem 5.1 to the theory of one-parameter semigroups of operators are discussed. In particular a version of the Hille-Yosida generation theorem is deduced for ${\displaystyle C_0}$ semigroups ${\displaystyle {\left(S_t\right)}_{t\in \overline{{ℝ}^{+}}}}$ such that ${\displaystyle {\supset }_{t\in \overline{{ℝ}^{+}}}{(\varkappa \left(t\right))}^{-1}\parallel S_t\parallel <\infty }$.