On a characteristic property of Weyl quantization

A condition is found under which a linear continuous mapping $W_1\colon L_2(\mathscr M)\to\hat{L_2}(H)$ of the space $L_2(\mathscr M)$ of generalized functionals on a phase space $\mathscr M$ into the set $\hat{L_2}(H)$ of Hilbert–Schmidt operators on a Fok space $H$ differs by only a numerical factor from Weyl quantization $W$.