Dragan S. Djordjević

On the generalized Drazin inverse and generalized resolvent

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C*-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.

Additive results for the generalized Drazin inverse

Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if A denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B) is computed in terms of A and B. Thus, recent results of R. E. Hartwig, G. Wang and Y. Wei are extended to infinite dimensional settings with simplified proof. 2000 Mathematics subject classification: 47A05, 47A55, 15A09.

Outer Generalized Inverses in Rings

ABSTRACT In this article we formulate basic results on outer generalized inverses of elements in rings. We characterize elements which have the same idempotents related to their particular outer generalized inverses and investigate positive generalized inverses in C*-algebras.

Further Results on the Reverse Order Law for Generalized Inverses

The reverse order rule

Immunoinflammatory Response in Critically Ill Patients: Severe Sepsis and/or Trauma

(AB)^\dag=B^\dag A^\dag

Cytokine profile in severe gram-positive and gram-negative abdominal sepsis

for the Moore-Penrose inverse is established in several equivalent forms. Results related to other generalized inverses are also proved.

Full-rank and determinantal representation of the Drazin inverse

Immunoinflammatory response in critically ill patients is very complex. This review explores some of the new elements of immunoinflammatory response in severe sepsis, tumor necrosis factor-alpha in severe acute pancreatitis as a clinical example of immune response in sepsis, immune response in severe trauma with or without secondary sepsis, and genetic aspects of host immuno-inflammatory response to various insults in critically ill patients.

Products of EP operators on Hilbert spaces

Sepsis is a principal cause of death in critical care units worldwide and consumes considerable healthcare resources. The aim of our study was to determine whether the early cytokine profile can discriminate between Gram-positive and Gram-negative bacteraemia (GPB and GNB, respectively) and to assess the prognostic value regarding outcome in critically ill patients with severe abdominal sepsis. The outcome measure was hospital mortality. Blood samples were obtained from 165 adult patients with confirmed severe abdominal sepsis. Levels of the proinflammatory mediators TNF-α, IL-8, IL-12 and IFN-γ and the anti-inflammatory mediators IL-1ra, IL-4, IL-10 and TGF-β1 were determined and correlated with the nature of the bacteria isolated from the blood culture and outcome. The cytokine profile in our study indicated that the TNF-α levels were 2-fold, IL-8 were 3.3-fold, IFN-γ were 13-fold, IL-1ra were 1.05-fold, IL-4 were 1.4-fold and IL-10 were 1.83-fold higher in the GNB group compared with the GPB group. The TNF-α levels were 4.7-fold, IL-8 were 4.6-fold, IL-1ra were 1.5-fold and IL-10 were 3.3-fold higher in the non-survivors compared with the survivors.

Star, left-star, and right-star partial orders in Rickart ∗-rings

Abstract In this article, we introduce a full-rank representation of the Drazin inverse A D of a given complex matrix A, which is based on an arbitrary full-rank decomposition of A l , l⩾k , where k is the index of A. Using this general representation, we introduce a determinantal representation of the Drazin inverse. More precisely, we represent elements of the Drazin inverse A D as a fraction of two expressions involving minors of the order rank (A k ) , k= ind (A) , taken from the matrices A and rank invariant powers A l , l⩾k . Also, we examine conditions for the existence of the Drazin inverse for matrices whose elements are taken from an integral domain. Finally, a few correlations between the minors of the Drazin inverse A D , powers of the Drazin inverse and the minors of the matrix A k , k= ind (A) , are explicitly derived.

Factorization of EP elements in C*-algebras

A Hilbert space operator A is called the EP operator if the range of A is equal to the range of its adjoint A*. In this article necessary and sufficient conditions are given for a product of two EP operators with closed ranges to be an EP operator with a closed range. Thus, a generalization of the well-known result of Hartwig and Katz (Linear Algebra Appl. 252 (1997), 339-345) is given.