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, an annihilation operator $a$ has the complex plane as spectrum (and coherent states as eigenstates), while the creation operator $a^*$, its adjoint, has empty spectrum. Here $a$ and $a^*$ act on a Hilbert space with a fixed countable orthonormal basis written as $|n\rangle$ ($n=0,1,2,,\ldots$) as 
$$a|n\rangle=\sqrt{n}|n-1\rangle,~~~a^*|n\rangle=\sqrt{n+1}|n+1\rangle.$$
(The terminology is as in https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator.) 

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