Abstract:

In this paper, the adjoint operator problem of the commutators $[A, B] = AB-BA$ of two unbounded operators $A$ and $B$ is studied by the spectral theory and block operator matrix theory. The sufficient conditions for the relationship $[A,B]^*=(AB-BA)^*=B^*A^*-A^*B^*=-[A^*,B^*]$ hold are given. In the end, concrete examples is given to illustrate the effectiveness of criterions.

Key words: Commutators, Adjoint operator, Spectrum