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 ${\mathcal{C}}_{\hat{I}}^{\circ }¦(\mathcal{H})$-valued functions. We will study weakly${}^{\ast }$ integrable $\mathcal{B}(\mathcal{H})$-valued functions and establish sufficient conditions for such functions to be Pettis integrable as ${\mathcal{C}}_{\hat{I}}^{\circ }¦(\mathcal{H})$-valued functions. In addition, we prove the inequality $${\Vert {\int }_E{\mathcal{A}}^{\ast }\mathcal{B}d\hat{I}¼\Vert }_{\hat{I}{¦}^{(p)}}⩽\Vert \mathcal{A}{\Vert }_{L_s^q}\cdot {\Vert \sqrt[p]{{\int }_E|\mathcal{B}{|}^{p}d\hat{I}¼}\Vert }_{\hat{I}{¦}^{(p)}},$$ where $\hat{I}{¦}^{(p)}$ is $p$-modification of the function $\hat{I}¦$ and the functions $\mathcal{A}$ and $\mathcal{B}$ belong to the suitable spaces of operator-valued functions. Finally, under some additional integrability assumptions on $\mathcal{B}$ we provide similar estimates of the Pettis norm.