Invariant subspaces and Deddens algebras

Abstract

It is shown that if the Deddens algebra DT associated with a quasinilpotent operator T on a complex Banach space is closed and localizing then T has a nontrivial closed hyperinvariant subspace.Ministerio de Economía y CompetitividadJunta de Andalucí

On the submultiplicativity and subadditivity of the spectral and essential spectral radius

“…In this article, we find other less restrictive assumptions under which these inequalities hold. Closely related conditions to our assumptions have already appeared in different contexts (see, e.g., [13], [12], [15], [8], and the references therein). The paper is organized as follows.…”

“…Note that the conditions ac = ca, bc = cb, and ab = cba, where c is powerbounded, are natural. Operators satisfying closely related conditions have already been studied in the context of spectral algebras and related invariant subspace problems (see, e.g., [13,Proposition 2.3], [12], [15], [8], and the references therein).…”

Volterra-type operators on the minimal Möbius-invariant space

“…The study of the Deddens algebra was originally introduced by Deddens [14], where he assumed that A is an invertible operator and . Later, it received the attention of many scholars (see [14, 19, 20, 34–37, 42, 43]). Recently, Petrovic and Sievewright [37] studied the Deddens algebra associated with compact composition operators on Hardy spaces, where A is not necessarily invertible, and they have demonstrated that the operators and belong to the Deddens algebra .…”

Invariant subspace of composition operators on Hardy space

“…The advantage about our proposed definition is that A can not be an invertible operator. For details and facts about Deddens algebra, we refer the reader to [10,13,15,16,19]. Multiplication operator M h is defined to be M h f (z) = h(z)f (z) for h ∈ H ∞ (D) and f ∈ H p .…”