On the Dual Representation of the Congruence Kernels and the Related Delsarte Type Transmutations of Multidimensional Differential Operators

We analyze a dual representation of the congruence operator kernels subject to a pair of multidimensional differential operators on a Hilbert space within both the spectral operator theory and Volterrian projector operator calculus. The related Volterrian type factorization of Fredholm operators is effectively solved for the case of special trace class elliptic operators. Based on the existence of operator kernels, spectrally congruent to the related pairs of operators, they are constructed by means of two approaches, using both of the kernel representation of the spectral operators in related Hilbert spaces and of the factorization property of Fredholm operators within the Volterrian projector operator calculus. In case of the multidimensional trace-class operator valued algebra of pseudo-differential expressions we stated that the corresponding kernel of a Fredholm operator, factorized by means of Volterrian kernel operators and commuting to a given elliptic representative, coincides with its fractional power. Application to construction of the related Delsarte type transmutation operators is presented.