Singular Differential Equations and g-Drazin Invertible Operators

We extend results of Favini, Nashed, and Zhao on singular differential equations using the g-Drazin inverse and the order of a quasinilpotent operator in the sense of Miekka and Nevanlinna. Two classes of singularly perturbed differential equations are studied using the continuity properties of the g-Drazin inverse obtained by Koliha and Rakočević.

Polaroid Operators with Svep and Perturbations of Property (Gaw)

In this paper we establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (gaw) holds. In this work, we consider commutative perturbations by algebraic operator and quasinilpotent operator for T ∈ B(X ) such that T * satisfies property (gaw). We prove that if A is an algebraic and T ∈ PS(X ) is such that AT = TA, then ƒ(T * + A*) satisfies property (gaw) for every ƒ ∈ Hc(σ(T + A)). Moreover, we show that if Q is a quasi-nilpotent operator and T ∈ PS(X ) is such that TQ = QT, then ƒ(T * + Q*) satisfies the property (gaw) for every ƒ ∈ Hc(σ(T +Q)). At the end of this paper, we apply the obtained results to a number of subclasses of PS(X ).

A decomposition theorem in II1-factors

Building on results of Haagerup and Schultz, we decompose an arbitrary operator in a diffuse, finite von Neumann algebra into the sum of a normal operator and an s.o.t.-quasinilpotent operator. We also prove an analogue of Weyl's inequality relating eigenvalues and singular values for operators in a diffuse, finite von Neumann algebra.

A NON-QUASINILPOTENT OPERATOR WHICH COMMUTES WITH A QUASINILPOTENT SHIFT

We will use a result of Bade, Dales and Laursen to construct a quasinilpotent weighted shift and an operator commuting with it with spectrum more than one point.

Diagonals of nilpotent operators

The purpose of the present note is to answer the following question of T. A. Gillespie,learned from G. J. Murphy [4]: for which sequences{an} of complex numbers does there exist a quasinilpotent operator Q on a (separable, infinite-dimensional, complex) Hilbert space H, which has{an} as a diagonal, that is (Qen,en)=n for some orthonormal basis{en} in H?

Invariant Complements to Closed Invariant Subspaces

The question under what conditions a closed invariant subspace possesses a closed invariant complement is of major importance in operator theory. In general it remains unanswered. In this paper we drop the requirement that the invariant complement be closed. We show in section 1 that the question is answerable under fairly mild conditions for a quasinilpotent operator (Theorem 1.5). These conditions will cover the case of a quasinilpotent operator with dense range and no point spectrum. In section 2 we discuss the consequences for the Volterra operator V. Since V is unicellular, its proper closed invariant subspaces do not possess closed invariant complements. However, they are all algebraically complemented (Proposition 2.1).