Extended eigenvalues and the Volterra operator

Abstract

Abstract. In this paper we consider the integral Volterra operator on the space L 2 ð0; 1Þ. We say that a complex number is an extended eigenvalue of V if there exists a nonzero operator X satisfying the equation XV ¼ VX. We show that the set of extended eigenvalues of V is precisely the interval ð0; 1Þ and the corresponding eigenvectors may be chosen to be integral operators as well.

On The Extended Eigenvalues of Some Volterra Operators

“…Nevertheless, the question whether B K = {K} for each operator in this class remains open. In this paper we generalize some results of [3] to a subclass of K 0 . In particular, we show that operators of this class possess nontrivial extended eigenvalues.…”

“…In particular, we show that operators of this class possess nontrivial extended eigenvalues. In Section 2 we show that the extended eigenvectors can be selected to generalize [3,Theorem 7]. Namely, it was shown there that, for λ > 1, if φ(x) = x/λ then the composition operator C φ is an extended eigenvector for V 0 (the simple…”

Extended eigenvalues for Cesàro operators

“…They also showed when dim H < ∞ that the set of extended eigenvalues for T ∈ B(H) reduces to {1} if and only if σ(T ) = {α} for some complex number α = 0. Finally, an example was given by Shkarin [22] of a compact quasinilpotent operator on a Hilbert space whose set of extended eigenvalues reduces to {1}, answering at once two questions raised by Biswas, Lambert and the third author [2].…”

SPECTRAL RADIUS ALGEBRAS AND C0 CONTRACTIONS. II

“…A major role in these results was played by the ideal Q A = {T : R m TR −1 m → 0}. We state the facts that are used in this paper and direct the reader to the articles [2][3][4][5][6][7][8][9] for more information. PROPOSITION 1.1.…”

“…Finally, B A = L(H) if and only if the operator A is similar to a constant multiple of an isometry. 92 S. Petrovic [2] A contraction A is completely nonunitary if there is no invariant subspace M for A such that A| M is a unitary operator. A completely nonunitary contraction A is said to be of class C 0 if there exists a nonzero function h ∈ H ∞ such that h(A) = 0.…”