Some properties on extended eigenvalues and extended eigenvectors

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...

On the Extended Eigenvalues of Some Volterra 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...

Notes on Eigenvalues and Eigenvectors

Exercise 4. Let λ be an eigenvalue of A and let Eλ(A) = {x ∈ C|Ax = λx} denote the set of all eigenvectors of A associated with λ (including the zero vector, which is not really considered an eigenvector). Show that this set is a (nontrivial) subspace of C. Definition 5. Given A ∈ Cm×m, the function pm(λ) = det(λI − A) is a polynomial of degree at most m. This polynomial is called the character...

Hyperinvariant subspaces and extended eigenvalues

An extended eigenvalue for an operator A is a scalar λ for which the operator equation AX = λXA has a nonzero solution. Several scenarios are investigated where the existence of non-unimodular extended eigenvalues leads to invariant or hyperinvariant subspaces. For a bounded operator A on a complex Hilbert space H, the set EE(A) of extended eigenvalues for A is defined to be the set of those co...