In a finite-dimensional Hilbert space, the spectrum of the adjoint $A^*$ of an operator $A$ is the complex conjugate of the spectrum of $A$.

But in infinite dimensions this need no longer be the case. For example, in Fock space, an annihilation operator $a$ has the complex plane as spectrum, while its adjoint, the creation operator $a^*$ has empty spectrum.

What is the general relation between the spectrum of an operator densely defined on A Hilbert space and that of its adjoint?