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

In this paper we will investigate Pettis integrability of $\mathcal{C}^{\circ}_{\Phi}(\mathcal{H})$-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 $\mathcal{C}^{\circ}_{\Phi}(\mathcal{H})$-valued functions. In addition, we prove the inequality $$\left\|\int_E\mathscr{A}^*\mathscr{B}d\mu \right\|_{\Phi^{(p)}} \leqslant \|\mathscr{A}\|_{L^q_s}\cdot\left\|\sqrt[p]{\int_E|\mathscr{B}|^pd\mu}\right\|_{\Phi^{(p)}},$$ where $\Phi^{(p)}$ is $p$-modification of the function $\Phi$ 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.