In this paper, we give an explicit formula for the Moore-Penrose INVERSE of W, denoted by W y, on L ( ). As an application, we give a characterization for some operator classes that are weaker than p-hyponormal with W y. Moreover, we give speci c examples illustrating these classes.

A new Schulz-type method to compute the Moore-Penrose INVERSE of a matrix is proposed. Every iteration of the method involves four matrix multiplications. It is proved that this method converge with fourth-order. A wide set of numerical comparisons shows that the average number of matrix multiplications and the average CPU time of our method are considerably less than those of other methods. For each of sizes n×n and n×(n+10), n=100, 200, 300, 400, ten random matrices were chosen to make these comparisons.

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We will introduce and study several algebraic properties of the covariance cosets in C*-algebras. Indeed, we will characterize the covariance coset in terms of commutators. Also, we will show that for an invertible element b, the covariance coset of b-1 coincides with the covariance coset of b*. Moreover, if b is normal, then the covariance coset of b coincides with the covariance coset of b*. In addition, we will prove that the covariance coset is a cone.

Laparoscopy has made a revolution in surgical procedures and treatment of various diseases but its complications are still under investigation. Intra-abdominal visceral and vessel injuries, trocar site hernia, and leaving foreign bodies into the peritoneal cavity are among some laparoscopic surgery complications. This is a rare report of Penrose drain migration following incomplete laparoscopic Fundoplication surgery. The patient was a 47- year- old woman, who was a candidate for Touplet Fundoplication via laparoscopic approach due to refractory gastro-esophageal reflux disease (GERD). While wrapping a Penrose drain around the esophagus, the patient had a cardiorespiratory arrest. Attempts to remove the Penrose drain were unsuccessful and the surgical procedure was terminated due to patient's condition. Four months later, after a long period of dysphagia and abdominal pain, the Penrose drain was defecated via rectum.

An element of Banach algebra is EP, if it commutes with its Moore-Penrose INVERSE. We present a number of new characterizations of EP elements in Banach algebra.

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We consider the perturbation and expression for the generalized INVERSE and Moore-Penrose INVERSE of closed linear operator under a weaker perturbation condition. As a application, we also investigate the perturbation for the Moore-Penrose INVERSE of closed EP operator. Some new and interesting perturbation results and examples are obtained in this paper.           

In this paper, we  nd explicit solution to the operator equation TXS 􀀀 SX T = A in the general setting of the adjointable operators between Hilbert C-modules, when T; S have closed ranges and S is a self adjoint operator.