Rank One Perturbations of Diagonal Operators Without Eigenvalues

Abstract

In this paper, we prove that every diagonal operator on a Hilbert space of which is of multiplicity one and has perfect spectrum admits a rank one perturbation without eigenvalues. This answers a question of Ionascu.Mathematics Subject Classification. 47A10, 47B06, 47A75, 47B15.

Weighted Join Operators on Directed Trees

“…Certainly, one could have defined the rank one extension of W u , and otherwise, these are bounded finite rank perturbations of diagonal operators. Needless to say, the later class has been studied extensively in the literature (refer, for example, to [67,35,25,26,27,24,42,46]). Also, the way in which W f,g is defined (cf.…”

“…We formally introduce these conditions below (cf. [35, Proposition 2.4(iv)], [46,Proposition 4.1]). Definition 4.5.…”

“…16. By [46,Theorem 7.1], for any diagonal operator D with simple point spectrum and perfect spectrum, there exists a bounded rank one perturbation h ⊗ k of arbitrarily small positive norm such that D + h ⊗ k has no point spectrum. This situation does not appear in the context of rank one extensions of weighted join operators, where the point spectrum is always non-empty.…”

“…The class of rank one perturbations of diagonal operators has been studied extensively in the context of hyperinvariant subspace problem [67,35,25,26,27,24,42,46] and spectral analysis [71,30,8,9,22,10]. The reader is referred to [63] for a survey on the classical theory of self-adjoint rank one perturbations of self-adjoint operators (refer also to [47] for its connection with the theory of singular integral operators).…”

Spectral dissection of finite rank perturbations of normal operators

“…In [30], Klaja proves that if N is diagonalizable and its spectrum is a perfect set, then N possesses a rank one perturbation without eigenvalues. If N is compact and self-adjoint, this is no longer true in general; the criterion for the existence of rank one perturbation of this type was given in [4].…”