The Spectrum of Annihilation Plus Orthogonal Creation

Abstract

Let ℌ be a Hilbert space and F(ℌ) be the full Fock space generated by ℌ. For v ∈ ℌ, the (left) creation operator l(v) : F(ℌ) → F(ℌ),f↦ v ⊗ f, and its adjoint, the (left) annihilation operator l(v)^*, are defined. If v, w ∈ℌ are orthonormal, it is proved that the spectrum of the operator l(v) + l(w)^* is purely continuous and conincides with the closed unit disk.