Extended spectrum and extended eigenspaces of quasinormal operators

Abstract

We say that a complex number $\lambda$ is an extended eigenvalue of a bounded linear operator $T$ on a Hilbert space $\mathcal{H}$ if there exists a nonzero bounded linear operator $X$ acting on $\mathcal{H}$, called the extended eigenvector associated to $\lambda$, and satisfying the equation $TX=\lambda XT$. In this article, we describe the sets of extended eigenvalues and extended eigenvectors for the quasinormal operators.