Soient $E$, $F$ des espaces de Banach $L_{\varphi }$, $L_{\psi }$ des espaces d’Orlicz, on définit les applications $\varphi -\psi$ sommantes de $E$ dans $F$. On montre que de telles applications sont $\varphi -\psi$ radonifiantes de $E$ dans $\sigma (F^{\text{'}\text{'}},F^{\text{'}})$.On donne une factorisation caractéristique des applications $\varphi -0$ sommantes.

Let ϰ be a positive, continuous, submultiplicative function on ${ℝ}^{+}$ such that $li{m}_{t\to \infty }{e}^{-\omega t}{t}^{-\alpha }\varkappa \left(t\right)=a$ for some ω ∈ ℝ, α ∈ $\overline{{ℝ}^{+}}$ and $a\in {ℝ}^{+}$. For every λ ∈ (ω,∞) let ${\varphi }_{\lambda }\left(t\right)=e^{-\lambda t}$ for $t\in {ℝ}^{+}$. Let $L_{\varkappa }^1\left({ℝ}^{+}\right)$ be the space of functions Lebesgue integrable on ${ℝ}^{+}$ with weight $\varkappa$, and let E be a Banach space. Consider the map ${\varphi }_{•}:(\omega ,\infty )\ni \lambda \to {\varphi }_{\lambda }\in L_{\varkappa }^1\left({ℝ}^{+}\right)$. Theorem 5.1 of the present paper characterizes the range of the linear map $T\to T{\varphi }_{•}$ defined on $L(L_{\varkappa }^1\left({ℝ}^{+}\right);E)$, generalizing a result established by B. Hennig and F. Neubrander for $\varkappa \left(t\right)=e^{\omega t}$. If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder’s characterization...