Operators constructed by means of basic sequences and nuclear matrices

AbstractIn this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space $\ell _{1}$ℓ1 of all absolutely summable sequences. Examples of nuclear operators over the space $\ell _{1}$ℓ1 are given and used to construct operators over general Banach spaces with specific approximation numbers.

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On Spaces of Compact Operators in Non-Archimedean Banach Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1989-065-x ◽

1989 ◽

Vol 32 (4) ◽

pp. 450-458

Author(s):

Takemitsu Kiyosawa

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Convergent Sequence ◽

Compact Operators ◽

Linear Operators ◽

Continuous Linear ◽

Infinite Dimensional ◽

Valued Field ◽

Infinite Dimensional Banach Space ◽

Continuous Linear Operators

AbstractLet K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.

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Approximation numbers of nuclear operators and geometry of Banach spaces

Archiv der Mathematik ◽

10.1007/bf01190003 ◽

1991 ◽

Vol 57 (2) ◽

pp. 155-168 ◽

Cited By ~ 4

Author(s):

Albrecht Pietsch

Keyword(s):

Banach Spaces ◽

Approximation Numbers ◽

Nuclear Operators

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The Bishop-Phelps-Bollobàs Property for Compact Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-2016-036-6 ◽

2018 ◽

Vol 70 (1) ◽

pp. 53-57 ◽

Cited By ~ 5

Author(s):

Sheldon Dantas ◽

Domingo García ◽

Manuel Maestre ◽

Miguel Martín

Keyword(s):

Hilbert Space ◽

Topological Space ◽

Banach Spaces ◽

Positive Measure ◽

Compact Operators ◽

Function Spaces ◽

Sequence Spaces ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Nikodym Property

AbstractWe study the Bishop-Phelps-Bollobàs property (BPBp) for compact operators. We present some abstract techniques that allow us to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let X and Y be Banach spaces. If (c0, Y) has the BPBp for compact operators, then so do (C0(L), Y) for every locally compactHausdorò topological space L and (X, Y) whenever X* is isometrically isomorphic to . If X* has the Radon-Nikodým property and (X), Y) has the BPBp for compact operators, then so does (L1(μ, X), Y) for every positive measure μ; as a consequence, (L1(μ, X), Y) has the BPBp for compact operators when X and Y are finite-dimensional or Y is a Hilbert space and X = c0 or X = Lp(v) for any positive measure v and 1 < p < ∞. For , if (X, (Y)) has the BPBp for compact operators, then so does (X, Lp(μ, Y)) for every positive measure μ such that L1(μ) is infinite-dimensional. If (X, Y) has the BPBp for compact operators, then so do (X, L∞(μ, Y)) for every σ-finite positive measure μ and (X, C(K, Y)) for every compact Hausdorff topological space K.

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Uniformly factoring weakly compact operators and parametrised dualisation

Forum of Mathematics Sigma ◽

10.1017/fms.2020.68 ◽

2021 ◽

Vol 9 ◽

Author(s):

L. Antunes ◽

K. Beanland ◽

B. M. Braga

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Borel Subset ◽

Compact Operators ◽

Schauder Basis ◽

Bounded Operators ◽

Weakly Compact ◽

Weakly Compact Operator ◽

Borel Map ◽

Weakly Compact Operators

Abstract This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

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On a Characterization of Convergence in Banach Spaces with a Schauder Basis

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2021/1640183 ◽

2021 ◽

Vol 2021 ◽

pp. 1-5

Author(s):

Marat V. Markin ◽

Olivia B. Soghomonian

Keyword(s):

Banach Space ◽

Hilbert Space ◽

Banach Spaces ◽

Separable Hilbert Space ◽

Linear Operators ◽

Schauder Basis ◽

Infinite Dimensional ◽

Summable Sequence ◽

Convergent Sequences

We extend the well-known characterizations of convergence in the spaces l p ( 1 ≤ p < ∞ ) of p -summable sequence and c 0 of vanishing sequences to a general characterization of convergence in a Banach space with a Schauder basis and obtain as instant corollaries characterizations of convergence in an infinite-dimensional separable Hilbert space and the space c of convergent sequences.“The method in the present paper is abstract and is phrased in terms of Banach spaces, linear operators, and so on. This has the advantage of greater simplicity in proof and greater generality in applications.” Jacob T. Schwartz

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Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽

10.3390/axioms10030150 ◽

2021 ◽

Vol 10 (3) ◽

pp. 150

Author(s):

Andriy Zagorodnyuk ◽

Anna Hihliuk

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Analytic Functions ◽

Entire Functions ◽

Dimensional Algebra ◽

Dimensional Banach Space ◽

Infinite Dimensional ◽

Linear Subspaces ◽

Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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A modified Krasnosellkii-Mann algorithms for computing fixed points of multivalued quasi-nonexpansive mappings in Banach spaces

Creative Mathematics and Informatics ◽

10.37193/cmi.2019.02.10 ◽

2019 ◽

Vol 28 (2) ◽

pp. 191-198

Author(s):

T. M. M. SOW

Keyword(s):

Strong Convergence ◽

Fixed Points ◽

Banach Spaces ◽

Iterative Scheme ◽

Nonexpansive Mappings ◽

Mann Iteration ◽

Infinite Dimensional ◽

Mann Algorithm

It is well known that Krasnoselskii-Mann iteration of nonexpansive mappings find application in many areas of mathematics and know to be weakly convergent in the infinite dimensional setting. In this paper, we introduce and study an explicit iterative scheme by a modified Krasnoselskii-Mann algorithm for approximating fixed points of multivalued quasi-nonexpansive mappings in Banach spaces. Strong convergence of the sequence generated by this algorithm is established. There is no compactness assumption. The results obtained in this paper are significant improvement on important recent results.

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