Some Conclusions About Hermitian Operators Extended to Non-Hermitian Cases

This paper extends some well-known conclusions about Hermitian operators in quantum mechanics to the case of non-Hermitian operators. Specifically, (1) it presents the eigenvalue structure of non-Hermitian operators satisfying algebraic equations; (2) for a class of diagonalizable non-Hermitian operators, it clarifies the relationship between commutativity, degenerate subspaces, and block diagonalization, and provides a visual representation; (3) taking the four-site spin-1/2 Heisenberg XXZ chain as an example, it specifically illustrates the physical implications and applications of these extended results.