Spherical Heinz transform of operator tuples

In this paper, we introduce the concept of the spherical Heinz transform for operator tuples, thereby extending the notion of the generalized mean transform in one-dimensional case to a multivariable operator setting. We study some of its properties and prove several inequalities involving the joint numerical radius and the joint operator norm of the spherical Aluthge and Heinz transforms. Moreover, several generalizations and refinements of some fundamental inequalities are also obtained.