In this paper we study the pseudo-spectra of the rotated harmonic oscillator. The pseudo-spectra of an operator are subsets in the complex plane which describe where the resolvent is large in norm. The study of such subsets allows us to understand the stability of the spectrum under perturbations and the possible calculation of ‘false eigenvalues’ far from the spectrum by algorithms for computing eigenvalues. The rotated harmonic oscillator is the simplest classic non-self-adjoint quadratic Hamiltonian, which has already been studied by Davies and Boulton. In one of his works, Boulton states a conjecture about the pseudo-spectra of this operator, which describes the instabilities for high energies. We can deduce this conjecture from a study of Dencker, Sjöstrand and Zworski, which gives bounds on the resolvent for a semi-classical pseudo-differential operator in a very general setting. In the present paper, we give a more elementary proof of this result using only some non-trivial localization scheme in the frequency variable.

Extensions of the concept of asymptotic independence which are useful in the study of abelian C$^*$-algebras generated by the union of two of their C$^*$-subalgebras are examined. Non-trivial examples are also given in terms of classes of bounded continuous functions defined on a locally compact group.