Banach Sequence Spaces Research Articles

The Brown–Halmos theorem for discrete Wiener–Hopf operators

We prove an analogue of the Brown–Halmos theorem for discrete Wiener–Hopf operators acting on separable rearrangement-invariant Banach sequence spaces.

Pathology of submeasures and Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideals

We address some phenomena about the interaction between lower semicontinuous submeasures on N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {N}}$$\\end{document} and Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideals. We give a partial answers to the question of whether every nonpathological tall Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideals using sequences in Banach spaces.

Narrow operators on [formula omitted]-complete lattice-normed spaces

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C-complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C-compact orthogonally additive operator T:X→Y from an atomless C-complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator TN:E(X)→E(X) to be C-compact. We also show that the sum of two orthogonally additive operators G and T, where G is narrow and T is laterally-to-norm continuous and C-compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space CE.

The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces

Abstract We calculate the essential norm of bounded diagonal infinite matrices acting on Köthe sequence spaces. As a consequence of our result, we obtain a recent criteria for the compactness of multiplication operator acting on Köthe sequence spaces.

Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper.

Woven (Weaving) Frames in Banach Spaces

Banach frames are defined by the straightforward generalization of Hilbert space frames. Woven (weaving) frames are the recent generalization of standard frames which appeared in the applications of distributed signal processing. In this paper, we introduce the concepts of woven (weaving) Bessel and frame sequences in Banach spaces and characterize the woven frames in terms of bounded operators. We also give some equivalent conditions for woven Xd-frame in Banach spaces.

ON THE EXISTENCE OF SOLUTIONS OF INFINITE SYSTEM OF HADAMARD-TYPE FRACTIONAL FUNCTIONAL INTEGRAL EQUATION IN THE BANACH SEQUENCE SPACE LP, P >1 AND COUPLED SEMI-ANALYTICAL METHOD TO SOLVE IT.

This article presents a study where the tool measure of noncompactness (MNC) and Meir-Keeler condensing operator (MKCO) are employed to establish the existence of solutions to an infinite system of Hadamard-type fractional functional integral equation in the Banach sequence space lp, p > 1. Additionally, an illustrative example is provided to validate the obtained results. Furthermore, we employ a coupled semianalytical method based on the modified homotopy perturbation method (mHPM) and Adomian’s decomposition method (ADM), to approximate the solution for the given example.

New Banach Sequence Spaces Defined by Jordan Totient Function

In this study, a special lower triangular matrix derived by combining Riesz matrix and Jordan totient matrix is used to construct new Banach spaces. $\alpha-$,$\beta-$,$\gamma-$duals of the resulting spaces are obtained and some matrix operators are characterized.

On Subsequential Averages of Sequences in Banach Spaces

On Subsequential Averages of Sequences in Banach Spaces

Some Results on Sequences in Banach Spaces

Some Results on Sequences in Banach Spaces

Spectral properties of generalized Cesàro operators in sequence spaces

The generalized Cesàro operators C_t, for tin [0,1], were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}, such as ell ^p, c_0, c, bv_0, bv and, as recently shown in Curbera et al. (J Math Anal Appl 507:31, 2022) [26], also in the discrete Cesàro spaces ces(p) and their (isomorphic) dual spaces d_p. In most cases C_t (tnot =1) is compact and its spectra and point spectrum, together with the corresponding eigenspaces, are known. We study these properties of C_t, as well as their linear dynamics and mean ergodicity, when they act in certain non-normable sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}. Besides {{mathbb {C}}}^{{{mathbb {N}}}_0} itself, the Fréchet spaces considered are ell (p+), ces(p+) and d(p+), for 1le p<infty , as well as the (LB)-spaces ell (p-), ces(p-) and d(p-), for 1<ple infty .

Symmetric Banach sequence spaces respect Weyl submajorization

Let E E be a symmetric Banach sequence space. We show that there exists an equivalent symmetric norm on E E which is monotone with respect to the Weyl (i.e., logarithmic) submajorization. Surprisingly, this purely commutative result is proved by a very non-commutative method.

On the transfer of convergence between two sequences in Banach spaces

"Let $(X,\|\cdot\|)$ be a Banach space and $T:A\to X$ a contraction mapping, where $A\subset X$ is a closed set. Consider a sequence $\{x_n\}\subset A$ and define the sequence $\{y_n\}\subset X$, by $y_n=x_n+T\left(x_{\sigma(n)}\right)$, where $\{\sigma(n)\}$ is a sequence of natural numbers. We highlight some general conditions so that the two sequences $\{x_n\}$ and $\{y_n\}$ are simultaneously convergent. Both cases: 1) $\sigma(n)&lt;n$, for all $n$, and 2) $\sigma(n)\ge n$, for all $n$, are discussed. In the first case, a general Picard iteration procedure is inferred. The results are then extended to sequences of mappings and some appropriate applications are also proposed."

Barbashin type characterizations for the uniform polynomial stability and instability of evolution families

Abstract The aim of this paper is to give some discrete and continuous versions of Barbashin type theorems for the uniform polynomial stability and instability of evolution families in Banach spaces, by using Banach sequence spaces in ℋ ⁢ ( ℕ ≥ 1 ) {\mathcal{H}(\mathbb{N}_{\geq 1})} and Banach function spaces in ℋ ⁢ ( ℝ ≥ 1 ) {\mathcal{H}(\mathbb{R}_{\geq 1})} , respectively. Variants for uniform polynomial stability and instability of some well-known results in the exponential stability theory and polynomial stability theory are obtained.

The generalized Banach sequence spaces and their dual spaces based on permutation symmetric gauge norms

In the present paper, we introduce and study a class of norms α on the sequence space , called permutation symmetric gauge norms, which properly contains the classical class of . For each , we define the generalized sequence space , which is proved to be a Banach space, and then we obtain a characterization of in terms of the projection onto . For the duality, the expected results in the classical sequence spaces () are still valid in these new settings, including the characterization of dual space and the existence of Hölder's inequality. It is worthy pointing out that the generalized sequence space is not a reflexive space, which is different from the usual sequence spaces.

Existence of Solutions for Nonlinear Integral Equations in Tempered Sequence Spaces via Generalized Darbo-Type Theorem

Two concepts—one of Darbo-type theorem and the other of Banach sequence spaces—play a very important and active role in ongoing research on existence problems. We first demonstrate the generalized Darbo-type fixed point theorems involving the concept of continuous functions. Keeping one of these theorems into our account, we study the existence of solutions of system of nonlinear integral equations in the setting of tempered sequence space. Moreover, a very interesting and illustrative example is designed to visualize our findings.

On New Banach Sequence Spaces Involving Leonardo Numbers and the Associated Mapping Ideal

In the present study, we have constructed new Banach sequence spaces ℓ p L , c 0 L , c L , and ℓ ∞ L , where L = l v , k is a regular matrix defined by l v , k = l k / l v + 2 − v + 2 , 0 ≤ k ≤ v , 0 , k &gt; v , for all v , k = 0 , 1 , 2 , ⋯ , where l = l k is a sequence of Leonardo numbers. We study their topological and inclusion relations and construct Schauder bases of the sequence spaces ℓ p L , c 0 L , and c L . Besides, α -, β - and γ -duals of the aforementioned spaces are computed. We state and prove results of the characterization of the matrix classes between the sequence spaces ℓ p L , c 0 L , c L , and ℓ ∞ L to any one of the spaces ℓ 1 , c 0 , c , and ℓ ∞ . Finally, under a definite functional ρ and a weighted sequence of positive reals r , we introduce new sequence spaces c 0 L , r ρ and ℓ p L , r ρ . We present some geometric and topological properties of these spaces, as well as the eigenvalue distribution of ideal mappings generated by these spaces and s -numbers.

Surjective isometries on Banach sequence spaces: A survey

Abstract In this survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries for various Banach spaces.

Meir-Keeler condensing operator to prove existence of solution for infinite systems of differential equations in the Banach space and numerical method to find the solution

In this paper, we establish the existence of solution for infinite systems of differential equations in the Banach sequence space n(?), ?p(1 ? p &lt; ?) and c by using Meier-Keeler condensing operators. With the help of examples we illustrate our results in the sequence spaces. Also for validity of the results, we find an approximation of solution by using a suitable method with high accuracy.

Existence of solution of infinite systems of singular integral equations of two variables in C(I × I, ℓp) with I = [0, T], T &gt; 0 and 1 &lt; p &lt; ∞ using Hausdorff measure of noncompactness

In this article, we discuss the solvability of infinite systems of singular integral equations of two variables in the Banach sequence spaces C(I ? I, ?p) with I = [0, T], T &gt; 0 and 1 &lt; p &lt; ? with the help of Meir-Keeler condensing operators and Hausdorff measure of noncompactness. With an example, we illustrate our findings.