Search

First, we give conditions on a Banach lattice E$E$ so that an operator T$T$ from E$E$ to any Banach space is disjoint p$p$‐convergent if and only if T$T$ is almost Dunford–Pettis. Then, we study when adjoints of positive operators between Banach lattices are disjoint p$p$‐convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices E$E$ and F$F$: (i) a positive operator T:E→F$T: E \rightarrow F$ is almost weak p$p$‐convergent if and only if T∗$T^*$ is disjoint p$p$‐convergent; (ii) E∗$E^*$ has order continuous norm or F∗$F^*$ has the positive Schur property of order p$p$. Very recent results are improved, examples are given and applications of the main results are provided. [ABSTRACT FROM AUTHOR]

In this paper, we characterize bounded, compact, and order bounded sums of weighted composition-differentiation operators from Bergman-type spaces to weighted Banach spaces of analytic functions. [ABSTRACT FROM AUTHOR]

This paper studies the basic topological properties of weighted composition operators on the weighted sequence spaces , with , given by a weight sequence of positive reals such as boundedness, compactness, compactness of differences of two operators, formulas for their essential norms, and description of closed range operators. Previously these properties were studied by Luan and Khoi in the case of Hilbert space . Their methods can be also applied with some minor modifications to the case of Banach spaces with . They based essentially on using the dual spaces of continuous linear functionals and, consequently, cannot be applied to the quasi-Banach case . Moreover, some of them do not work even in . Motivated by these reasons, we develop a more universal approach that allows studying the whole scale of spaces . To this end, we establish the necessary and sufficient conditions for a linear operator to be compact on an abstract quasi-Banach sequence space. These conditions are new even in the case of Banach spaces. Moreover, we introduce the new characteristic, the -essential norm of a continuous linear operator on a quasi-Banach space . This characteristic measures the distance in the operator metric, between and the set of all -compact operators on . Here an operator is -compact on if is compact and coordinatewise continuous on . We show that for with the essential and -essential norms of a weighted composition operator coincide, whereas for we do not know whether the same is true or not. Our main results for weighted composition operators in are as follows: We provide criteria for an operator to be bounded, compact, or closed range, and completely describe the pairs of operators with compact difference; as well as some exact formula for the -essential norm. Some key aspects of our approach can be used for other operators and scales of spaces. [ABSTRACT FROM AUTHOR]

In this paper we present necessary and sufficient conditions characterizing compact sets in the spaces of Nuclear and Hilbert-Schmidt operators on Hilbert spaces. Based on these results, we also characterize compact sets in the space of probability measures on separable Hilbert spaces. These results are then used in the study of optimal feedback control theory for stochastic systems. We prove existence of optimal feedback control laws for standard and several nonstandard control problems. Further, we present necessary conditions of optimality and an algorithm and related convergence theorem whereby optimal control laws can be constructed. We present also a result characterizing compact sets in the space of probability measures on separable Hilbert spaces. These results have interesting applications in control theory on infinite dimensional Banach spaces as presented in the last section. [ABSTRACT FROM AUTHOR]

In this work, we proposed some generalized Darbo type fixed point theorem by employing the concept of generalized operators and measure of non-compactness within the framework of partial order Banach spaces. Some fixed point theorems and their consequences are presented as part of our exploration. These theorems encompass the presentation of various fixed point theorems and the subsequent implications they entail. Further, we demonstrate the practical application of our findings by establishing the existence of solutions for delay differential equations. To substantiate our conclusions, we provide numerical estimations based on a real-world example. This research contributes to the advancement of fixed point theory in the context of partial order Banach spaces and showcases its practical relevance through a concrete application and empirical support. [ABSTRACT FROM AUTHOR]

Introduction A subset A of a Banach space X is called limited, if every W*-null sequence (xn*) in X* converges uniformly on A; that is, supa∈Ax∈A |< xn*. a >| → 0. Every relatively compact set is limited and if every limited subset of X is relatively compact, then X has the Gelfand-Phillips (GP) property. For example, every separable Banach space and every Schur space have the GP property [13]. If A ⊆ X* and every weakly null (resp. weakly null limited) sequence (xn) in X converges uniformly on A, we say that A is an L−set (resp. L−limited set). Each relatively weakly compact set in X* is an L−limited set and if the converse also holds, X has the L−limited property [3, 21]. A bounded linear operator T: X → Y between two Banach spaces is limited completely continuous if it carries limited weakly null sequences in X to norm null ones in Y. The class of all limited completely continuous operators from X to Y is denoted by Lcc(X.Y) [22]. A sequence (xn) in a Banach space X is called weakly p−summable with 1 ≤ p < ∞, if for each x* ∈ X*, the sequence (x* (xn)) ∈ lp and a sequence (xn) ⊆ X is said to be weakly p-convergent to x ∈ X if (xn − x) ∈ lpw(X), where lpw(X)denoted the space of all weakly p-summable sequences in X. The weakly ∞-convergent sequences are the weakly convergent sequences. A bounded subset A ⊆ X is relatively weakly p−compact if every sequence in A has a weakly p−convergent subsequence. If limit of each weakly p−convergent subsequence is in A, then that A is called a weakly p−compact set. A bounded linear operator T: X → Y between two Banach spaces is p−convergent (resp. limited p-convergent) if it carries weakly p−summable (resp. limited weakly p−summable) sequences in X to norm null ones in Y [7, 26]. Recently the concepts of p-Schur and limited p−Schur properties in Banach spaces are introduced. In fact, a Banach space X has the p−Schur (resp. limited p−Schur) property if all weakly p−compact (resp. limited weakly p−compact) subsets of X are relatively compact, or equivalently every sequence (xn) ∈ lpw(X) (resp. limited sequence (xn) ∈ lpw(X) is norm null [26, 8]. The aim of this paper is to study the class of almost L−limited sets of order p in dual Banach lattices and disjoint limited p-convergent operators. Also, we characterize Banach lattices in which two classes of almost L−limited sets of order p and L−limited sets of order p in their dual coincide. In particular, a positive answer to an open question posed in [21] is given. In fact, we show that although L−limited subsets of E* strictly contain w*-sequentially compact sets, but w*-sequentially compact subsets are relatively weakly compact if and only if L−limited subsets are relatively weakly compact. Moreover, some results of the disjoint limited p−Schur property as a generalization of the limited p−Schur property are investigated and some important consequences about this property are established. In this article we assume that 1 ≤ p < ∞, unless otherwise stated. We recall some definitions and notations from Banach lattice theory. The norm ||.|| of a Banach lattice E is order continuous if for each generalized net (xa) such that xa ↓ 0 in E, (xa) converges to 0 for the norm ||.||, where the notation xa ↓ 0 means that the net (xa) is decreasing, its infimum exists and infa a (xa) = 0. A subset A of E is called solid if |x| ≤ |y| for some y ∈ A implies that x ∈ A and the solid hull of A is the set sol(A) = {y ∈ E: |y| ≤ |x|, for some x ∈ A}. Throughout this article, X denotes a Banach space, X* refers to the dual of X, E denotes a Banach lattice, E+ = {x ∈ E ∶ x ≥ 0} is the positive cone of E and BE is the closed unit ball of E. A subset of a Banach lattice is called order bounded if it is contained in an order interval. For terminologies concerning Banach lattice theory we refer the reader to [1, 20]. Material and Methods In this paper the class of almost L−limited sets of order p in dual Banach lattices and disjoint limited p−convergent operators are studied. Also, Banach lattices in which two classes of almost L−limited sets of order p and L−limited sets of order p in their dual coincide, are characterized. In particular, a positive answer to an open question posed in [21] is given. Moreover, some results of the disjoint limited p−Schur property as a generalization of the limited p−Schur property are investigated and some important consequences about this property are obtained. Results and discussion The followings are the main results of our paper. Theorem. For a Banach lattice E, the following are equivalent: (a) E is a Grothendieck space, (b) E has the L− limited property, (c) E has the pL−limited property. Theorem. Let E be a σ-Dedekind complete Banach lattice. Then the following are equivalent: (a) E has the disjoint limited p−Schur property, (b) for each Banach space Y, Lpcd(E. Y) = L(E. Y), (c) Lpcd(E. l∞) = L(E. l∞). Theorem. Let E be a σ-Dedekind complete Banach lattice. Then E has the disjoint limited p−Schur property if and only if every disjoint limited positive sequence (xn) ∈ lpw(E) is norm null. Theorem. Let E be a σ-Dedekind complete Banach lattice with the type q (with 1 < q ≤ 2. p ≥ q'). Then the following are equivalent: (a) E has the p−wDP* property, (b) clpd(E. Y) = Cpd(E. Y), for each Banach space Y, (c) Clpd(E. c0) = Cpd(E. c0). 0 Theorem. For a Banach lattice E, the following are equivalent: (a) each almost pL−limited set in E* is a pL−limited set, (b) for each Banach space Y, Lpcd(E. Y) = Lpc(E.Y), (c) Lpcd(E, l∞) = Lpc(E, l∞). Conclusion The following conclusions are obtained from this research. Banach lattices in which almost pL−limited sets and pL−limited sets are relatively weakly compact are studied. As an application, we give a connection between the L−limited property and Grothendieck Banach lattices. Also, some results of the disjoint limited p−Schur property as a generalization of the limited p−Schur property are investigated and some important consequences about this property are established. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the essential pseudospectrum of bounded linear operator pencils is invariant under perturbation of each completely continuous linear operators on ultrametric Banach spaces over a spherically complete field K and we establish a characterization of the essential pseudospectrum of bounded linear operator pencils by means of the spectrum of each perturbed completely continuous operators. Furthermore, we introduce and study the notion of (n, ε)-pseudospectrum of bounded linear operators and the concept of (n, ε)-pseudospectrum of bounded linear operator pencils on ultrametric Banach spaces. We establish some results about them. Finally, several examples are provided. [ABSTRACT FROM AUTHOR]

In this work we study the Riemann-Liouville fractional integral of order α ∈ (0 , 1 / p) as an operator from L p (I ; X) into L q (I ; X) , with 1 ≤ q ≤ p / (1 - p α) , whether I = [ t 0 , t 1 ] or I = [ t 0 , ∞) and X is a Banach space. Our main result provides necessary and sufficient conditions to ensure the compactness of the Riemann-Liouville fractional integral from L p (t 0 , t 1 ; X) into L q (t 0 , t 1 ; X) , when 1 ≤ q < p / (1 - p α) . [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as C-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate the controllability in the α -norm of a coupled system of integrodifferential equations with state-dependent nonlocal conditions in generalized Banach spaces. We establish sufficient conditions for the system's controllability using resolvent operator theory introduced by Grimmer, fractional power operators, and fixed-point theorems associated with generalized measures of noncompactness for condensing operators in vector Banach spaces. Finally, we present an application example to validate the proposed methodology in this research. [ABSTRACT FROM AUTHOR]

Let X and Y be Banach spaces. We provide the representation of the dual space of compact operators K(X, Y) as a subspace of bounded linear operators L(X, Y). The main results are: (1) If Y is separable, then the dual forms of K(X, Y) can be represented by the integral operator and the elements of C[0, 1]. (2) If X** has the weak Radon-Nikodym property, then the dual forms of K(X, Y) can be represented by the trace of some tensor products. [ABSTRACT FROM AUTHOR]

We study the duality and norm completeness in the new classes of limitedly L-weakly compact and Dunford-Pettis L-weakly compact operators from Banach spaces to Banach lattices. [ABSTRACT FROM AUTHOR]

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators. [ABSTRACT FROM AUTHOR]

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced. [ABSTRACT FROM AUTHOR]

In the present paper we define the degree of nondensifiability (DND for short) of a bounded linear operator T on a Banach space and analyze its properties and relations with the Hausdorff measure of noncompactness (MNC for short) of T. As an application of our results, we have obtained a formula to find the essential spectral radius of a bounded operator T on a Banach space as well as we have provided the best possible lower bound for the Hyers-Ulam stability constant of T in terms of the aforementioned DND. [ABSTRACT FROM AUTHOR]

The paper deals with some further results concerning the class of two-Lipschitz operators. We prove first an isometric isomorphism identification of two-Lipschitz operators and Lipschitz operators. After defining and characterizing the adjoint of a two-Lipschitz operator, we prove a Schauder type theorem on the compactness of the adjoint. We study the extension of two-Lipschitz operators from the cartesian product of two complemented subspaces of a Banach space to the cartesian product of whole spaces. Also, we show that every two-Lipschitz functional defined on the cartesian product of two pointed metric spaces admits an extension with the same two-Lipschitz norm under some requirements on domaine spaces. [ABSTRACT FROM AUTHOR]

An operator in a Banach space is said to be recurrent if the set is dense in . The operator is said to be weakly sequentially recurrent if the set is weakly dense in . Costakis et al. [Complex Anal. Oper. Theory 8 (8), 1601–1643] ask if should be recurrent whenever so is . This question has been answered negatively by Grivaux et al. [arXiv: 2212.03652]. In this paper, we prove the existence of an operator weakly sequentially recurrent such that is not. [ABSTRACT FROM AUTHOR]

Let X, Y be two Banach spaces over K = ℝ or ℂ , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems. [ABSTRACT FROM AUTHOR]

In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification EC$E_{\mathbb {C}}$ of a real vector lattice E and introduce the lateral order on EC$E_{\mathbb {C}}$. Our first main result asserts that the set of all fragments Fv$\mathfrak {F}_v$ of an element v∈EC$v\in E_{\mathbb {C}}$ of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification EC$E_{\mathbb {C}}$ of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator T:EC→X$\mathcal {T}: E_{\mathbb {C}} \rightarrow X$ from the complexification EC$E_{\mathbb {C}}$ of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator T$\mathcal {T}$ from EC$E_{\mathbb {C}}$ to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from EC$E_{\mathbb {C}}$ to a complex Banach lattice (ℓp(D)C$(\ell _p(\mathcal {D})_{\mathbb {C}}$ is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators. [ABSTRACT FROM AUTHOR]

We give some characterizations of disjointly weakly compact sets in Banach lattices, namely, those sets in whose solid hulls every disjoint sequence converges weakly to zero. As an application, we prove that a bounded linear operator from a Banach space to a Banach lattice is an almost (L) limited operator if and only if it is a disjointly weakly compact operator, indeed, an operator which carries bounded sets to disjointly weakly compact ones. Some results on weak precompactness and (L -, M-)weak compactness of disjointly weakly compact operators are also obtained. [ABSTRACT FROM AUTHOR]

We prove that, given two Banach spaces X and Y and bounded, closed convex sets C ⊆ X and D ⊆ Y , if a nonzero element z ∈ co ¯ (C ⊗ D) ⊆ X ⊗ ^ π Y is a preserved extreme point then z = x 0 ⊗ y 0 for some preserved extreme points x 0 ∈ C and y 0 ∈ D , whenever K (X , Y ∗) separates points of X ⊗ ^ π Y (in particular, whenever X or Y has the compact approximation property). Moreover, we prove that if x 0 ∈ C and y 0 ∈ D are weak-strongly exposed points then x 0 ⊗ y 0 is weak-strongly exposed in co ¯ (C ⊗ D) whenever x 0 ⊗ y 0 has a neighbourhood system for the weak topology defined by compact operators. Furthermore, we find a Banach space X isomorphic to ℓ 2 with a weak-strongly exposed point x 0 ∈ B X such that x 0 ⊗ x 0 is not a weak-strongly exposed point of the unit ball of X ⊗ ^ π X . [ABSTRACT FROM AUTHOR]

In this paper, we study partial integral operators on Banach–Kantorovich spaces over a ring of measurable functions. We obtain a decomposition of the cyclic modular spectrum of a bounded modular linear operator on a Banach–Kantorovich space in the form of a measurable bundle of spectra of bounded operators on Banach spaces. The classical Banach spaces with mixed norm are endowed with the structure of Banach–Kantorovich modules. We use such representations to show that every partial integral operator on a space with a mixed norm can be represented as a measurable bundle of integral operators. In particular, we show the cyclic compactness of such operators and, as an application, prove the Fredholm -alternative. We also give an example of a partial integral operator with a nonempty cyclically modular discrete spectrum, while its modular discrete spectrum is an empty set. [ABSTRACT FROM AUTHOR]

Every k-homogeneous (continuous) polynomial between Banach spaces admits a unique Aron-Berner extension to the biduals. Our main result states that, for every σ-finite measure µ, every polynomial P ∈ with Y -valued Aron-Berner extension is representable, in other words, it factors through ℓ1 in the form P = A◦Q where Q is a polynomial and A is an operator. This may be viewed as a polynomial strengthening of the Dunford-Pettis-Phillips theorem stating that every weakly compact operator on L1(µ) is representable. We introduce the Radon-Nikodým polynomials and show that every polynomial with Y -valued Aron-Berner extension is Radon-Nikodým. Finally, we prove that every Radon-Nikodým polynomial is unconditionally converging. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we present some investigations on the class of almost (L) limited operators. We show that an operator T : X → E , from a Banach space X to a Banach lattice E, is almost (L) limited iff its adjoint carries disjoint almost L-sequences to norm null ones. In addition, we improve several results obtained by Oughajji et al. In its second part, we study the relationship between the class of weakly precompact operators and that of order weakly compact (resp. b-weakly compact) operators. Among other things, we show that for a Banach lattice E and a Banach space X the following statements are equivalent: Every order weakly compact (resp. b-weakly compact) operator T : E → X is weakly precompact; The norm of E ′ is order continuous or X does not contain any isomorphic copy of ℓ 1 . [ABSTRACT FROM AUTHOR]

In this research paper, we develop some aspects of the theory of Fredholm linear operators acting in p-adic (or non-Archimedean) Banach spaces. In this regard, we establish sufficient conditions for the p-adic Fredholmness of the algebraic sum of unbounded linear operators. Next, we study the perturbation of p-adic upper semi-Fredholm operators under strictly singular operators. Moreover, we give some results concerning quasi-compact and p-adic lower semi-Fredholm operators. [ABSTRACT FROM AUTHOR]

The paper is largely concerned with the possibility of obtaining a series representation for a compact linear map T acting between Banach spaces. It is known that, using the notions of j−$j-$eigenfunctions and j−$j-$ eigenvalues, such a representation is possible under certain conditions on T. Particular cases discussed include those in which T can be factorized through a Hilbert space, or has certain s‐numbers that are fast‐decaying. The notion of p‐compactness proves to be useful in this context; we give examples of maps that possess this property. [ABSTRACT FROM AUTHOR]

We prove lower bounds for the randomized approximation of the embedding ℓ 1 m ↪ ℓ ∞ m based on algorithms that use arbitrary linear (hence non-adaptive) information provided by a (randomized) measurement matrix N ∈ R n × m . These lower bounds reflect the increasing difficulty of the problem for m → ∞ , namely, a term log m in the complexity n. This result implies that non-compact operators between arbitrary Banach spaces are not approximable using non-adaptive Monte Carlo methods. We also compare these lower bounds for non-adaptive methods with upper bounds based on adaptive, randomized methods for recovery for which the complexity n only exhibits a (log log m) -dependence. In doing so we give an example of linear problems where the error for adaptive vs. non-adaptive Monte Carlo methods shows a gap of order n 1 / 2 (log n) − 1 / 2 . [ABSTRACT FROM AUTHOR]

For Banach spaces X, Y, we consider a distance problem in the space of bounded linear operators L (X , Y). Motivated by a recent article, we obtain sufficient conditions so that for a compact operator T ∈ L (X , Y) and a closed subspace Z ⊂ Y , the following equation holds, which relates global approximation with local approximation: d (T , L (X , Z)) = sup { d (T x , Z) : x ∈ X , ‖ x ‖ = 1 }. In some cases, we show that the supremum is attained at an extreme point of the corresponding unit ball. Furthermore, we obtain some situations when the following equivalence holds: T ⊥ B L (X , Z) ⇔ T * * x 0 * * ⊥ B Z ⊥ ⊥ ⇔ T * * ⊥ B L (X * * , Z ⊥ ⊥) , for some x 0 * * ∈ X * * satisfying ‖ T * * x 0 * * ‖ = ‖ T * * ‖ ‖ x 0 * * ‖ , where Z ⊥ is the annihilator of Z. One such situation is when Z is an L 1 − predual space and an M − ideal in Y and T is a multi-smooth operator of finite order. Another such situation is when X is an abstract L 1 − space and T is a multi-smooth operator of finite order. Finally, as a consequence of the results, we obtain a sufficient condition for proximinality of a subspace Z in Y. [ABSTRACT FROM AUTHOR]

In this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental operator, generated by the linear part, and on the nonlinear multivalued term. This purpose is achieved by combining a fixed point theorem, a selection theorem and a containment theorem. Further we provide another existence result in reflexive spaces by using the classical Hahn-Banach theorem and a new selection proposition, proved here, for a multimap guided by an operator. This setting allows us to remove some assumptions required in the previous existence theorem. As a consequence of this last result we obtain the controllability of a problem driven by a wave equation on which an appropriate perturbation acts. [ABSTRACT FROM AUTHOR]

In a normed space we characterize the approximate orthogonality in terms of extreme points of the closed unit ball in dual space. There is also another (and older) definition of an approximate orthogonality and therefore in this paper both approximate orthogonalities are compared. Moreover, we characterize the approximate orthogonality in the class of compact operators on a reflexive Banach space. [ABSTRACT FROM AUTHOR]

In this note we characterize the bounded compact approximation property via Calkin representations for Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the domain of a recently defined conservative matrix, constructed by means of the Schröder numbers in the spaces of p-absolutely summable sequences and bounded sequences. We determine the ß-duals of the Banach spaces, introduced here, and present characterization of some matrix operators. Moreover, we give the characterization of certain compact operators via the Hausdorff measure of noncompactness. [ABSTRACT FROM AUTHOR]

Let A be a Banach algebra. For f ∈ A*, we inspect the weak sequential properties of the well-known map Tf : A → A*, Tf (a) = fa, where fa ∈ A* is defined by fa(x) = f (ax) for all x ∈ A. We provide equivalent conditions for when Tf is completely continuous for every f ∈ A*, and for when Tf maps weakly precompact sets onto L-sets for every f ∈ A*. Our results have applications to the algebra of compact operators K(X) on a Banach space X. [ABSTRACT FROM AUTHOR]

For two Banach spaces X and Y, we denote by K(X,Y) (resp. B(X,Y)) the space of all compact (resp. bounded) linear operators from X to Y. In this paper, we show that for 1\leq p,q<\infty, K(\ell _{p},\ell _{q}) is an 8-Lipschitz retract of B(\ell _{p},\ell _{q}). [ABSTRACT FROM AUTHOR]

We study the interpolation properties of weakly compact bilinear operators by the real method and also by the complex method. We also study the factorization property of weakly compact bilinear operators through reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

The article is devoted to integral operators for nonlocally compact group modules. Operator valued functions for nonlocally compact groups and their normed spaces of different types are investigated. For this purpose quasi-invariant measures on nonlocally compact groups are used. Estimates of norms of integral operators are given. Moreover, convolutions of functions having operator values and values in Banach spaces are scrutinized. Examples are given. [ABSTRACT FROM AUTHOR]

In a more recent paper [23], Banach spaces of absolutely k -summable series |Tφ|k have been introduced and studied by Sarıgöl and Agarwal. In this paper, we characterize matrix and compact operators from the space |Tφ|k to the classical spaces c0, c and l∞, by determining identities or estimates for operator norms and the Hausdorff measure of noncompactness. [ABSTRACT FROM AUTHOR]

The main intention of this paper is to deal with the weak singularity of solutions to some nonlinear Hadamard type fractional differential systems (HTFDSs) which could be viewed as the generalization of classic Hadamard fractional settings. Resorting to the lower and upper solutions technique and constructing the compatible weighted Banach space, the completely continuous operator described by Hadamard type fractional versions as well as a sufficient criterion for the existence of blow-up solutions to some nonlinear HTFDSs is established. Additionally, the comparative analysis on the blow-up rate affected by critical system parameters is also presented, and several examples clearly illustrate the effectiveness and efficiency of the proposed results. [ABSTRACT FROM AUTHOR]

As a consequence of a factorization theorem, we establish conditions on the Banach spaces E and F so that the k-adjoint of every linear operator/homogeneous polynomial from E to F belongs a given operator ideal I . Applications are provided for operators/polynomials on ℓ 1 and for the ideal of weakly compact operators. [ABSTRACT FROM AUTHOR]

We provide a new extension of Pitt's theorem for compact operators between quasi-Banach lattices which permits to describe unconditional bases of finite direct sums of Banach spaces \mathbb {X}_{1}\oplus \dots \oplus \mathbb {X}_{n} as direct sums of unconditional bases of their summands. The general splitting principle we obtain yields, in particular, that if each \mathbb {X}_{i} has a unique unconditional basis (up to equivalence and permutation), then \mathbb {X}_{1}\oplus \cdots \oplus \mathbb {X}_{n} has a unique unconditional basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space \ell _2\oplus \mathcal {T}^{(2)} has a unique unconditional basis. [ABSTRACT FROM AUTHOR]

Let X be a real reflexive Banach space and X* be its dual space. Let G 1 and G 2 be open subsets of X such that G 2 ⊂ G 1, 0 ∈ G2, and G1 is bounded. Let L: X ⊃ D (L) → X* be a densely defined linear maximal monotone operator, A: X ⊃ D (A) → 2 X* be a maximal monotone and positively homogeneous operator of degree γ > 0, C: X ⊃ D (C) → X* be a bounded demicontinuous operator of type (S +) w.r.t. D (L), and T: G 1 → 2 X* be a compact and upper-semicontinuous operator whose values are closed and convex sets in X* . We first take L = 0 and establish the existence of nonzero solutions of Ax + Cx + Tx ∋ 0 in the set G1 \ G2. Secondly, we assume that A is bounded and establish the existence of nonzero solutions of Lx + Ax + Cx ∋ 0 in G1\G2. We remove the restrictions γ ∈ (0, 1] for Ax + Cx + Tx ∋ 0 and γ = 1 for Lx + Ax + Cx ∋ 0 from such existing results in the literature. We also present applications to elliptic and parabolic partial differential equations in general divergence form satisfying Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property. [ABSTRACT FROM AUTHOR]

In this paper, we present a general Banach space of absolutely k-summable series using a triangle matrix operator and prove that this is a BK-space isometrically isomorphic to the space ℓ k {\ell_{k}}. We also establish the α - {\alpha-} , β - {\beta-} , γ-duals and base of the new space. Finally, we qualify some matrix and compact operators on the new space making use of the Hausdorff measure of noncompactness. Our results include, as particular cases, a number of well-known results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate conditions on a pair of Banach lattices E and F that tells us when every positive almost L-weakly compact (resp. almost M- weakly compact) operator T : E → F is weakly compact. Also, we present some necessary conditions that tells us when every weakly compact operator T : E → F is almost M-weakly compact (resp. almost L-weakly compact). In particular, we will prove that if every weakly compact operator from a Banach lattice E into a Banach space X is almost L-weakly compact, then E is a KB-space or X has the Dunford-Pettis property and the norm of E is order continuous. [ABSTRACT FROM AUTHOR]

Let K (X , Y) be the space of compact operators. For a proximinal subspace Z ⊂ Y , this paper deals with the question, when does every Y -valued compact operator admit a Z -valued compact best approximation? For any reflexive Banach space X and for a L 1 -predual space Y , if Z ⊂ Y is a strongly proximinal subspace of finite codimension, we show that K (X , Z) is a proximinal subspace of K (X , Y) under an additional condition on the position of K (X , Z). When Y is a c 0 -direct sum of finite dimensional spaces we achieve a strong transitivity result by showing that for any proximinal subspace of finite codimension Z ⊂ Y , every Y -valued bounded operator admits a best Z -valued compact approximation. [ABSTRACT FROM AUTHOR]

We study best approximations to compact operators between Banach spaces and Hilbert spaces, from the point of view of Birkhoff-James orthogonality and semi-inner-products. As an application of the present study, some distance formulae are presented in the space of compact operators. The special case of bounded linear functionals as compact operators is treated separately and some applications to best approximations in reflexive, strictly convex and smooth Banach spaces are discussed. An explicit example is presented in ℓpn spaces, where 1 < p < ∞, to illustrate the applicability of the methods developed in this article. A comparative analysis of the results presented in this article with the well-known classical duality principle in approximation theory is conducted to demonstrate the advantage in the former case, from a computational point of view. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach spaces and let A , B : X → Y be two operators, where A is a linear homeomorphism and B is a C 1 -compact operator. Sufficient weakly coerciveness conditions are provided to assert that the perturbed operator A + B is a C 1 -diffeomorphism. The proof of our result is based on properties of Fredholm mappings as well as on local and global inverse mapping theorems. A corollary is provided. It shows that the operator B has one and only one fixed point if some weak coerciveness hypotheses are verified. As an application of our results, two examples are given for integral equations. [ABSTRACT FROM AUTHOR]

We extend the notions of joint and generalized spectral radii to cocycles acting on Banach spaces and obtain a version of Berger–Wang's formula when restricted to the space of cocycles taking values in the space of compact operators. Moreover, we observe that the previous quantities depends continuously on the underlying cocycle. [ABSTRACT FROM AUTHOR]

In this article, we prove the proximinality of closed unit ball of M-ideals of compact operators on Banach spaces. We show that every positive (self-adjoint) operator on a Hilbert space has a positive (self-adjoint) compact approximant from the closed unit ball of space of compact operators. We also show that K(ℓ1), the space of compact operators on ℓ1, is ball proximinal in B(ℓ1), the space of bounded operators on ℓ1, even though K(ℓ1) is not an M-ideal in B(ℓ1). Moreover, we prove the ball proximinality of M-embedded spaces in their biduals. [ABSTRACT FROM AUTHOR]