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Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.

In Chapter 16 we give applications to various topics of Banach space theory, the most spectacular of which is a sharpened version of a celebrated result of Jean Bourgain on the Λ p problem.

We study the Bishop-Phelps-Bollob\'as property for operators between Banach spaces. Sufficient conditions are given for generalized direct sums of Banach spaces with respect to a~uniformly monotone Banach sequence lattice to have the approximate hyperplane series property. This result implies that Bishop-Phelps-Bollob\'as theorem holds for operators from $\ell_1$ into such direct sums of Banach spaces. We also show that the direct sum of two spaces with the approximate hyperplane series property has such property whenever the norm of the direct sum is absolute., Comment: 26 pages

International audience; In a separable Banach space E, we study the invariance of a closed set K under the action of the evolution equation associated with a maximal dissipative linear operator A perturbed by a quasi-dissipative continuous term B. Using the distance to the closed set, we give a general necessary and sufficient condition for the invariance of K. Then, we apply our result to several examples of partial differential equations in Banach and Hilbert spaces.

A new geometric property of Banach spaces recently introduced by G. Kasparov and G. Yu, called Property (H), has important applications to the strong Novikov conjecture and the coarse Novikov conjecture, yet is far from being well understood. In this paper, we investigate various uniformly continuous maps on unit spheres of Banach spaces and prove that all separable Banach lattices with nontrivial cotype have Property (H).

We establish the conditions of bifurcation of the solutions of weakly perturbed operator equations in Banach spaces from the point 𝜀 = 0 and propose a convergent iterative procedure for finding the solutions in the form of segments of powers series in 𝜀 with pole at the point 𝜀 = 0.

Several new pre-orders are introduced and characterized on the set of all wg -Drazin invertible elements of a Banach algebra. We generalize recent results for rectangular matrices and bounded linear operators between Banach spaces, omitting some assumptions and adding new characterizations.

Let A be a complex Banach algebra, let be the set of all quasinilpotent elements in A and let be the set of all quasi-regular elements in A. We characterize generalized derivations g of A such that...

In this work, we introduce implicit and explicit iterations for solving the variational inclusion problem for the sum of two operators and the fixed point problem of nonexpansive mappings. We then prove its strong convergence theorems in the framework of Banach spaces. We finally provide some applications of the main results.

Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

A classical problem (initially studied by N. Kalton and A. Wilansky) concerns finding closed infinite dimensional subspaces of X / Y, where Y is a subspace of a Banach space X. We study the Banach lattice analogue of this question. For a Banach lattice X, we prove that X / Y contains a closed infinite dimensional sublattice under the following conditions: either (i) Y is a closed infinite codimensional subspace of X, and X is either order continuous or a C(K) space, where K is a compact subset of \({\mathbb {R}}^n\); or (ii) Y is the range of a compact operator.

The objective in this study is to use generalized iterative procedures in order to approximate solutions of an equation on a Banach space setting. In particular, we present a semi-local convergence analysis for these methods. Some applications are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type.

We propose an approach to the construction of solutions and quasisolutions of a boundary-value problem for the Lyapunov equation in a Banach space. If the necessary and sufficient conditions for the solvability of this boundary-value problem are satisfied, then the corresponding solutions of the problem are constructed by using the generalized inverse operator. As an example, we consider the problem in the space of bounded sequences with countably dimensional matrices.

A family of local atoms in a Banach space has been introduced and it has been generalized to an atomic system for operators in Banach spaces, which has been further led to introduce new frames for operators by Dastourian and Janfada, by making use of semi-inner products. Unlike the traditional way of considering sequences in the dual space, sequences in the original space are considered to study them. Appropriate changes have been made in the definitions of atomic systems and frames for operators to fit them for sequences in the dual space without using semi-inner products so that the new notion for Banach spaces can be thought of as a generalization of Banach frames. With some crucial assumptions, we show that frames for operators in Banach spaces share nice properties of frames for operators in Hilbert spaces.

Let${\mathcal C}[{\mathcal X}]$be any class of operators on a Banach space${\mathcal X}$, and let${Holo}^{-1}({\mathcal C})$denote the class of operatorsAfor which there exists a holomorphic functionfon a neighbourhood${\mathcal N}$of the spectrum σ(A) ofAsuch thatfis non-constant on connected components of${\mathcal N}$andf(A) lies in${\mathcal C}$. Let${{\mathcal R}[{\mathcal X}]}$denote the class of Riesz operators in${{\mathcal B}[{\mathcal X}]}$. This paper considers perturbation of operators$A\in\Phi_{+}({\mathcal X})\Cup\Phi_{-}({\mathcal X})$(the class of all upper or lower [semi] Fredholm operators) by commuting operators in$B\in{Holo}^{-1}({\mathcal R}[{\mathcal X}])$. We prove (amongst other results) that if πB(B) = ∏mi= 1(B− μi) is Riesz, then there exist decompositions${\mathcal X}=\oplus_{i=1}^m{{\mathcal X}_i}$and$B=\oplus_{i=1}^m{B|_{{\mathcal X}_i}}=\oplus_{i=1}^m{B_i}$such that: (i) If λ ≠ 0, then$\pi_B(A,\lambda)=\prod_{i=1}^m{(A-\lambda\mu_i)^{\alpha_i}} \in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) if and only if$A-\lambda B_0-\lambda\mu_i\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$), and (ii) (case λ = 0)$A\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) if and only if$A-B_0\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$), whereB0= ⊕mi= 1(Bi− μi); (iii) if$\pi_B(A,\lambda)\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) for some λ ≠ 0, then$A-\lambda B\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$).

We give conditions for the linear span of the positive L-weakly compact (resp. M-weakly compact) operators to be a Banach lattice under the regular norm, for that Banach lattice to have an order continuous norm, to be an AL-space or an AM-space. Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [2219]; Research Foundation of Namik Kemal UniversityNamik Kemal University [NKUBAP.00.10.AR.15.04] The first author was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) within the context of 2219-Post Doctoral Fellowship Program and by the Research Foundation of Namik Kemal University (Project No. NKUBAP.00.10.AR.15.04.).

For Banach algebras A and B, we show that if U=A×B is unital and commutative, each bi-Jordan homomorphism from U into a semisimple commutative Banach algebra D is a bihomomorphism.

In this paper, we introduce the concept of a new nonlinear mapping called demigeneralized in a Banach space. Then, using the shrinking projection method, we prove a strong convergence theorem for finding a common fixed point for a family of the new nonlinear mappings in a Banach space. We apply this result to obtain new strong convergence theorems in a Hilbert space and a Banach space, respectively.

In this paper, we introduce the notion of Saphar linear relation in a Banach space and we study the behaviour of such notion in polynomials.

We exhibit a new class of Banach spaces Y such that the pair ( c 0 , Y ) has the Bishop–Phelps–Bollobas property for operators. This class contains uniformly convex Banach spaces and spaces with the property β of Lindenstrauss. We also provide new examples of spaces in this class.

In this paper, sufficient conditions for the existence of a common fixed point for a Banach operator pair of mappings satisfying generalized contractive conditions in the frame work of CAT(0) spaces are obtained. As an application, related results on best approximation are derived. Our results generalize various known results in contemporary literature

In this paper, we propose an extension of quasi-equilibrium problems from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. The proposed problem is called quasi-variational problem. We study the convergence of some algorithms to solutions of the proposed nonconvex problems in Banach spaces.

In this study, we introduce the norm-attaining property for a dual pair of Banach spaces. We use this property to determine a quotient Banach space as a dual space. We also apply this property to obtain another proof of the James's characterization theorem regarding the reflexivity of a separable Banach space. In addition, the norm-attaining property of a Fourier space is studied.

In three previous papers, we discussed quasidense multifunctions from a Banach space into its dual, or, equivalently, quasidense subsets of the product of a Banach space and its dual. In this paper, we survey (without proofs) some of the main results about quasidensity, and give some simple limiting examples in Hilbert spaces, reflexive Banach spaces, and nonreflexive Banach spaces.

Let M be a Stein Banach manifold modeled on a separable Banach space with the bounded approximation property, Γ → M a (locally trivial) holomorphic Banach Lie group bundle over M, and O Γ → M ( C Γ → M ) the sheaf of germs of holomorphic (continuous) sections of Γ. We show that the natural map H 1 ( M , O Γ ) → H 1 ( M , C Γ ) of first cohomology sets is bijective.

The purpose of this paper is to introduce and study the split equality variational inclusion problems in the setting of Banach spaces. For solving this kind of problems, some new iterative algorithms are proposed. Under suitable conditions, some strong convergence theorems for the sequences generated by the proposed algorithm are proved. As applications, we shall utilize the results presented in the paper to study the split equality feasibility problems in Banach spaces and the split equality equilibrium problem in Banach spaces. The results presented in the paper are new.

In this paper we will study the real character on order Banach algebra with identity, and study character onorder Banach algebrawithout identity. We proved some properties on real character it. We define set of all character and prove it with some condition convex set and used it to prove all character is extreme point, we introduce dual cone in order Banach space .Also we show any order Banach algebra satisfying certain conditions is isomorphic to the space valued continuous function C0(x) forsuitable a locally compact Housdorf space.

We give a characterization of weakly subgaussian random elements that are γ-subgaussian in infinite-dimensional Banach and Hilbert spaces.

In this paper, we characterize the perturbations of the Moore–Penrose metric generalized inverse of closed operator in Banach spaces. Under the condition $R(\delta T)\subset R(T)$ , $N(T)\subset N(\delta T)$ , respectively, we get some new results about upper-bound estimates of $\|\bar{T}^{M}\|$ and $\|\bar{T}^{M}-T^{M}\|$ .

This paper is divided into four parts. In the first we study the existence of vector-valued Banach limits and show that a real Banach space with a monotone Schauder basis admits vector-valued Banach limits if and only if it is -complemented in its bidual. In the second we prove two vector-valued versions of Lorentz' intrinsic characterization of almost convergence. In the third we show that the unit sphere in the space of all continuous linear operators from to which are invariant under the shift operator on cannot be obtained via compositions of surjective linear isometries with vector-valued Banach limits. In the final part we show that if enjoys the Krein-Milman property, then the set of vector-valued Banach limits is a face of the unit ball in the space of all continuous linear operators from to which are invariant under the shift operator on .

Recently, Lafforgue introduced a new strengthening of Banach property (T), which he called strong Banach property (T) and showed that this property has implications regarding fixed point properties and Banach expanders. In this paper, we introduce a new strengthening of Banach property (T), called “robust Banach property (T)”, which is weaker than strong Banach property (T), but is still strong enough to ensure similar applications. Using the method of averaged projections in Banach spaces and introducing a new notion of angles between projections, we establish a criterion for robust Banach property (T) and show several examples of groups in which this criterion is fulfilled. We also derive several applications regarding fixed point properties and Banach expanders and give examples of these applications.

Our main result demonstrates that rather weak forms of the extendable local reflexivity and of the principle of local reflexivity are needed for the lifting of bounded convex approximation properties from Banach spaces to their dual spaces. This provides a unified approach to the lifting of various bounded approximation properties, including, besides the classical ones, the approximation property for pairs of Banach spaces, and the positive approximation property of Banach lattices.

In this paper, first we introduce some aspects of approximate fixed points, such as approximate fixed point property for subsets of a fuzzy Banach space and approximate eigenvalues of a self mapping on a fuzzy normed space. Then as their consequences, we prove more results for Fixed Point Theory on Fuzzy Banach and Banach Spaces.