This paper explores refinements of some operator norm inequalities through the generalized spherical Aluthge transform and the spherical Heinz transform. We introduce the spherical Schatten $p$-norm for operator tuples and establish several related inequalities. Additionally, equality conditions for some of these inequalities are also presented. Furthermore, we define the (joint) Schatten $p$-numerical radius and the Schatten hypo-$p$-norm for operator tuples, deriving some fundamental inequalities in this setting.
Categories: math.FA