scholarly journals Banach spaces for which the space of operators has 2𝔠 closed ideals

2021 ◽  
Vol 9 ◽  
Author(s):  
Daniel Freeman ◽  
Thomas Schlumprecht ◽  
András Zsák
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sequence Spaces ◽  
Type Spaces ◽  
The Continuum ◽  
General Conditions

Abstract We formulate general conditions which imply that ${\mathcal L}(X,Y)$ , the space of operators from a Banach space X to a Banach space Y, has $2^{{\mathfrak {c}}}$ closed ideals, where ${\mathfrak {c}}$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in ${\mathcal L}\left (\ell _p\oplus \ell _q\right )$ is exactly $2^{{\mathfrak {c}}}$ for all $1<p<q<\infty $ .

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

On the Construction of Sequence Spaces that have Schauder Bases

1966 ◽  
Vol 18 ◽  
pp. 1281-1293 ◽  
Author(s):  
William Ruckle
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sequence Spaces ◽  
Schauder Basis ◽  
Close Connection ◽  
Original Source ◽  
Schauder Bases ◽  
Bk Spaces ◽  
Made In

It is known that every Banach space which possesses a Schauder basis is essentially a space of sequences (6, Section 11.4). The primary objectives of this paper are: (1) to illustrate the close connection between sectionally bounded BK spaces and Banach spaces which have a Schauder basis, and (2) to consider some results in these theories in such a way as to render them easy and natural. In order to reach the largest number of readers we shall use (6) as the sole basis of our discussion. References to other authors are made in order to direct the reader to the original source of a theorem or to a related discussion.

scholarly journals On Antichains of Spreading Models of Banach Spaces

2010 ◽  
Vol 53 (1) ◽  
pp. 64-76
Author(s):  
Pandelis Dodos
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
The Continuum

AbstractWe show that for every separable Banach spaceX, either SPw(X) (the set of all spreading models ofXgenerated by weakly-null sequences inX, modulo equivalence) is countable, or SPw(X) contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell, and B. Sari.

scholarly journals Some remarks on regular Banach spaces

1996 ◽  
Vol 38 (2) ◽  
pp. 243-248 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Weakly Compact ◽  
Bounded Linear ◽  
Type Spaces

A Banach space E is said to be regular if every bounded linear operator from E into E′ is weakly compact. This property was studied in [7, 9] under the name Property (w). In [7], using James type spaces as constructed in [4], examples were given of regular Banach spaces which fail to have weakly sequentially complete duals, answering a question raised in [9]. In this paper, we present some more results concerning the regularity of James type spaces.

scholarly journals Kuelbs–Steadman Spaces for Banach Space-Valued Measures

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1005
Author(s):  
Antonio Boccuto ◽  
Bipan Hazarika ◽  
Hemanta Kalita
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Banach Spaces ◽  
Norm Convergence ◽  
Type Spaces ◽  
Additive Measures

We introduce Kuelbs–Steadman-type spaces ( K S p spaces) for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate the main properties and embeddings of L q -type spaces into K S p spaces, considering both the norm associated with the norm convergence of the involved integrals and that related to the weak convergence of the integrals.

scholarly journals On a theorem of Sobczyk

1991 ◽  
Vol 43 (1) ◽  
pp. 123-130 ◽  
Author(s):  
Aníbal Moltó
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Compact Space ◽  
Weak Topology ◽  
Topological Property ◽  
Continuum Hypothesis ◽  
Rosenthal Compact ◽  
Resolution Of The Identity ◽  
Corson Compact ◽  
The Continuum

In this paper the result of Sobczyk about complemented copies of c0 is extended to a class of Banach spaces X such that the unit ball of their dual endowed with the weak* topology has a certain topological property satisfied by every Corson-compact space. By means of a simple example it is shown that if Corson-compact is replaced by Rosenthal-compact, this extension does not hold. This example gives an easy proof of a result of Phillips and an easy solution to a question of Sobczyk about the existence of a Banach space E, c0 ⊂ E ⊂ l∞, such that E is not complemented in l∞ and c0 is not complemented in E. Assuming the continuum hypothesis, it is proved that there exists a Rosenthal-compact space K such that C(K) has no projectional resolution of the identity.

scholarly journals Matrix transformations involving certain Banach space valued sequence spaces

2000 ◽  
Vol 31 (2) ◽  
pp. 85-100
Author(s):  
J. K. Srivastava ◽  
B. K. Srivastava
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Operators ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Matrix Classes ◽  
The Matrix ◽  
Matrix Class

In this paper for Banach spaces $X$ and $Y$ we characterize matrix classes $ (\Gamma (X,\lambda)$, $ l_\infty(Y,\mu))$, $ (\Gamma(X,\lambda),C(Y,\mu))$, $ (\Gamma(X,\lambda)$, $ c_0(Y,\mu))$, $ (\Gamma(X,\lambda)$, $ \Gamma^*(Y,\mu))$, $ (l_1(X,\lambda)$, $ \Gamma(Y,\mu))$ and $ (c_0(X,\lambda)$, $ c_0(Y,\mu))$ of bounded linear operators involving $ X$- and $ Y$-valued sequence spaces. Further as an application of the matrix class $ (c_0(X,\lambda)$, $ c_0(Y,\mu))$ we investigate the Banach space $ B(c_0(X,\lambda)$, $ c_0(Y,\mu))$ of all bounded linear mappings of $ c_0(x,\lambda)$ to $ c_0(Y,\mu)$.

Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

1981 ◽  
Vol 90 (1-2) ◽  
pp. 63-70 ◽  
Author(s):  
Bernd Carl
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Besov Spaces ◽  
Eigenvalue Problems ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Entropy Numbers ◽  
Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

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

Axioms ◽  
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.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

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