  The aim of this work is to derive a symbol calculus on $L^2(\mathbb{R}^n)$
for multidimensional Hausdorff operators. Two aspects of this activity result
in two almost independent parts. While throughout the perturbation matrices are
supposed to be self-adjoint and form a commuting family, in the second part
they are additionally assumed to be positive definite. What relates these two
parts is the powerful method of diagonalization of a normal Hausdorff operator
elaborated earlier by the second author.
