The unique transformation matrix T in the block-diagonalization of a Hermitian matrix H, \(\tilde H=T^+HT\), is considered. The unitary matrix that brings a Hermitian matrix H into block-diagonal form can be uniquely determined under simple and transparent conditions. The block- diagonalization problem is studied in the framework of the second quantization so that the study starts from the corresponding operator relation \(\tilde H^\wedge= \hat T^+\hat H\hat T\). The operator \(\hat H\) has a well-defined matrix representation in any n-particle Fock space.
The existence of a block-diagonalization operator is demonstrated. This operator is unique and its construction is given. In the particular case of an operator \(\hat H\) given by a one-particle operator the block- diagonalization operator can be given in explicit form. Application to Green function theory is discussed. As an example various propagators are analyzed which describe the process of removal of one particle from the ground state of the system.