Hyperinvariant subspaces for some operators on locally convex spaces

Abstract

Abstract. Some results concerning hyperinvariant subspaces of some operators on locally convex spaces are considered. Denseness of a class of operators which have a hyperinvariant subspace in the algebra of locally bounded operators is proved.Keywords and phrases. Locally convex space, hyperinvariant subspace, operator.2000 Mathematics Subject Classification. Primary 47A15, 47B99, 46A32.1. Introduction. Let X be a locally convex space over the complex field C. Each system of seminorms P inducing its topology will be called a calibration. We denote by ᏼ(X) the collection of all calibrations on X and by Λ(X) all continuous seminorms with respect to the given topology. Let us denote by ᏸ(X) the set of all linear continuous operators on X and by (X) the set of compact operators on X, i.e., T ∈ (X) if there exists a neighborhood U such that T (U) is a relatively compact set. We shall denote by im(T ) the range of T and by ker T the null space of T . For a given P ∈ ᏼ(X), let P = {p α : α ∈ ∆}, where ∆ is some index set. Choose any α ∈ ∆ and let X α := X/ ker(p α ) denote the quotient space which is a normed space with respect to the norm x α α = p α (x) where x α = x + ker(p α ). The completeness of X α we denote byX α . It is well known that for the dual spaces the following relation holds X = {(X α ) , α ∈ ∆} (Floret and Wloka [2]). Let B be an absolutely convex and bounded set, then X B := {nB, n ∈ N} is a normed subspace in X with respect to the norm x B := inf{λ > 0 : x ∈ λB}. It is easy to see that for any p α ∈ P there is some λ α ≥ 0 such that

Quasi-Nilpotent Operators on Locally Convex Spaces

“…which implies that λ 1 ∈ σ a (Q P , T ) ⊂ σ(Q P , T ) (see [11]). This contradicts the supposition we made, so lemma is proved.…”