The spectrum of annihilation plus orthogonal creation (Q5933727)

Let \(f\) ang \(g\) be orthogonal unit vectors in a Hilbert space \(H\) and let \(\ell(f)\) and \(\ell(g)\) be the corresponding creation (or left multiplication) operators on the full Fock space \(F(H)\) over \(H\). The author uses elementary methods of the theory of linear operators to show that the spectrum of \(\ell(f)+\ell(g)^*\) is a disc. In the mean time more general results have been obtained in the so-called free probability theory (by means of much more advanced methods.).