Pettis integrability of functions with values in separable symmetrically-normed ideals and related norm estimates

Abstract

In this paper we will investigate Pettis integrability of $\mathcal{C}^{\circ}_Φ(\mathcal{H})$-valued functions. We will study weakly$^*$ integrable $\mathcal{B}(\mathcal{H})$-valued functions and establish sufficient conditions for such functions to be Pettis integrable as $\mathcal{C}^{\circ}_Φ(\mathcal{H})$-valued functions. In addition, we prove the inequality $$\left\|\int_E\mathscr{A}^*\mathscr{B}dμ\right\|_{Φ^{(p)}} \leqslant \|\mathscr{A}\|_{L^q_s}\cdot\left\|\sqrt[p]{\int_E|\mathscr{B}|^pdμ}\right\|_{Φ^{(p)}},$$ where $Φ^{(p)}$ is $p$-modification of the function $Φ$ and the functions $\mathscr{A}$ and $\mathscr{B}$ belong to the suitable spaces of operator-valued functions. Finally, under some additional integrability assumptions on $\mathscr{B}$ we provide similar estimates of the Pettis norm.