Extended spectrum, extended eigenspaces and normal operators

We say that a complex number λ is an extended eigenvalue of a bounded linear operator T on a Hilbert space H if there exists a nonzero bounded linear operator X acting on H , called extended eigenvector associated to λ, and satisfying the equation T X = λ X T . In this paper we describe the sets of extended eigenvalues and extended eigenvectors for the product of a positive and a self-adjoint operator which are both injective. We also treat the case of normal operators.