On a unitary transform in the space $L_2(\Omega,\mathbb R^3)$ connected with the Weyl decomposition

In the papers devoted to the dynamical inverse problem for the Naxwell system, in the framework of the BC-method, a unitary transform $M$: "solenoidal field$\to$transversal field" was introduced. In this paper $M$ is complemented by a transform $N$: "potential field$\to$longitudinal field." Isometry and completeness of $N$ are established. The transform $U=M\oplus N$ mentioned in the title, turns out to be a unitary oprator.