Abstract:
EP Banach space operators and EP Banach algebra elements are characterized using different kinds of factorizations. The results obtained generalize well-known characterizations of EP matrices, EP Hilbert space operators and EP $C^*$-algebra elements. Furthermore, new results that hold in these contexts are presented.

Abstract:
Weighted EP Banach space operators and Banach algebra elements are characterized using different kinds of factorizations. The results presented extend well-known characterizations of (weighted) EP matrices, (weighted) EP Hilbert space operators and (weighted) EP $C^*$-algebra elements.

Abstract:
Several characterizations of EP and normal Moore-Penrose invertible Banach algebra elements will be considered. The Banach space operator case will be also studied. The results of the present article will extend well known facts obtained in the frames of matrices and Hilbert space operators.

Abstract:
Let €:E ￠ ’X and :F ￠ ’X be bundles of Banach spaces, where X is a compact Hausdorff space, and let V be a Banach space. Let “( €) denote the space of sections of the bundle €. We obtain two representations of integral operators T: “( €) ￠ ’V in terms of measures. The first generalizes a recent result of P. Saab, the second generalizes a theorem of Grothendieck. We also study integral operators T: “( €) ￠ ’ “( ) which are C(X)-linear.

Abstract:
Given a Banach Algebra $A$ and $a\in A$, several relations among the Drazin spectrum of $a$ and the Drazin spectra of the multiplication operators $L_a$ and $R_a$ will be stated. The Banach space operator case will be also examined. Furthermore, a characterization of the Drazin spectrum will be considered.

Abstract:
A bounded linear operator which has a finite index and which is defined on a Banach space is often referred to in the literature as a Fredholm operator. Fredholm operators are important for a variety of reasons, one being the role that their index plays in global analysis. The aim of this paper is to prove the spectral theorem for compact operators in refined form and to describe some properties of the essential spectrum of general bounded operators by the use of the theorem of Fredholm operators. For this, we have analysed the Fredholm operator which is defined in a Banach space for some special characterisations.

Abstract:
By a result of Johnson, the Banach space $F=(\bigoplus_{n=1}^\infty \ell_1^n)_{\ell_\infty}$ contains a complemented copy of $\ell_1$. We identify $F$ with a complemented subspace of the space of (bounded, linear) operators on the reflexive space $(\bigoplus_{n=1}^\infty \ell_1^n)_{\ell_p}$ ($p\in (1,\infty))$, thus giving a negative answer to the problem posed in the monograph of Diestel and Uhl which asks whether the space of operators on a reflexive Banach space is Grothendieck.

Abstract:
Using continuous selections, we establish some existence results about the zeros of weakly continuous operators from a paracompact topological space into the dual of a reflexive real Banach space.

Abstract:
Using continuous selections, we establish some existence results about the zeros of weakly continuous operators from a paracompact topological space into the dual of a reflexive real Banach space.

Abstract:
We first introduce a modified proximal point algorithm formaximal monotone operators in a Banach space. Next, we obtain astrong convergence theorem for resolvents of maximal monotoneoperators in a Banach space which generalizes the previous resultby Kamimura and Takahashi in a Hilbert space. Using this result,we deal with the convex minimization problem and the variationalinequality problem in a Banach space.