Extended eigenvalues of composition operators

On Extended Eigenvalues of Operators

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λXA. We characterize the the set of extended eigenvalues for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C0 contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. We derive some appli...

On the Extended Eigenvalues of Some Volterra Operators

Intertwining Relations and Extended Eigenvalues for Analytic Toeplitz Operators

We study the intertwining relation XTφ = TψX where Tφ and Tψ are the Toeplitz operators induced on the Hardy space H 2 by analytic functions φ and ψ, bounded on the open unit disc U, and X is a nonzero bounded linear operator on H. Our work centers on the connection between intertwining and the image containment ψ(U) ⊂ φ(U), as well as on the nature of the intertwining operator X. We use our re...

Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V on the Wiener algebra W (D) of analytic functions on the unit discD of the complex plane C. A complex number is called an extended eigenvalue of V if there exists a nonzero operator A satisfying the equation AV = V A. We prove that the set of all extended eigenvalues of V is precisely the set C\{0}, and describe in terms of Du...

Eigenvalues of Operators with Gaps

This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential.