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 HH if there exists a nonzero bounded linear operator X   acting on HH, called extended eigenvector associated to λ  , and satisfying the equation TX=λXTTX=λXT. 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.