$U$-projection for the adjoint representation of the group $\mathrm{GL}(n,K)$

In the paper we study rings and fields of invariants for the adjoint representation of the group $\mathrm{GL}(n,K)$ over the field of zero characteristic. The aim of this paper is to construct the special linear operator, we call it $U$-projector, that maps any polynomial on the matrix algebra to an $U$-invariant rational function. In this paper we present two different constructions of $U$-projector. Using the $U$-projector we obtain the system of generators of the field of $U$-invariants of the adjoint representation of the group $\mathrm{GL}(n,K)$. We obtain the system of generators of the field of $U$-invariants for the restriction of the adjoint representation to the subgroup of block-diagonal matrices.