CLASSES OF SEQUENTIALLY LIMITED OPERATORS

2015 ◽  
Vol 58 (3) ◽  
pp. 573-586
Author(s):  
JAN H. FOURIE ◽  
ELROY D. ZEEKOEI
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Normed Space ◽  
Operator Ideal ◽  
Operator Ideals ◽  
Ideal Norm ◽  
Relevant Operator

AbstractThe purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.

Duality of the distance to closed operator ideals

2003 ◽  
Vol 133 (1) ◽  
pp. 197-212 ◽  
Author(s):  
Hans-Olav Tylli
Keyword(s):  
Banach Spaces ◽  
Closed Operator ◽  
Operator Ideal ◽  
Linear Operators ◽  
Distance Functions ◽  
Approximation Properties ◽  
Operator Ideals ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Special Operator

Special operator-ideal approximation properties (APs) of Banach spaces are employed to solve the problem of whether the distance functions S ↦ dist(S*, I(F*, E*)) and S ↦ dist(S, I*(E, F)) are uniformly comparable in each space L(E, F) of bounded linear operators. Here, I*(E, F) = {S ∈ L(E, F) : S* ∈ I(F*, E*)} stands for the adjoint ideal of the closed operator ideal I for Banach spaces E and F. Counterexamples are obtained for many classical surjective or injective Banach operator ideals I by solving two resulting ‘asymmetry’ problems for these operator-ideal APs.

scholarly journals The ideal of Lipschitz classical p-compact operators and its injective hull

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Toufik Tiaiba ◽  
Dahmane Achour
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Operator Ideal ◽  
Linear Case ◽  
Injective Hull ◽  
Nuclear Operators ◽  
The Ideal

Abstract We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

scholarly journals Complex interpolation of a Banach space with its dual

2000 ◽  
Vol 87 (2) ◽  
pp. 200
Author(s):  
Frédérique Watbled
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Vector Space ◽  
Unconditional Basis ◽  
Scalar Product ◽  
Dual Space ◽  
Function Spaces ◽  
Complex Interpolation

Let $X$ be a Banach space compatible with its antidual $\overline{X^*}$, where $\overline{X^*}$ stands for the vector space $X^*$ where the multiplication by a scalar is replaced by the multiplication $\lambda \odot x^* = \overline{\lambda} x^*$. Let $H$ be a Hilbert space intermediate between $X$ and $\overline{X^*}$ with a scalar product compatible with the duality $(X,X^*)$, and such that $X \cap \overline{X^*}$ is dense in $H$. Let $F$ denote the closure of $X \cap \overline{X^*}$ in $\overline{X^*}$ and suppose $X \cap \overline{X^*}$ is dense in $X$. Let $K$ denote the natural map which sends $H$ into the dual of $X \cap F$ and for every Banach space $A$ which contains $X \cap F$ densely let $A'$ be the realization of the dual space of $A$ inside the dual of $X \cap F$. We show that if $\vert \langle K^{-1}a, K^{-1}b \rangle_H \vert \leq \parallel a \parallel_{X'} \parallel b \parallel_{F'}$ whenever $a$ and $b$ are both in $X' \cap F'$ then $(X, \overline{X^*})_{\frac12} = H$ with equality of norms. In particular this equality holds true if $X$ embeds in $H$ or $H$ embeds densely in $X$. As other particular cases we mention spaces $X$ with a $1$-unconditional basis and Köthe function spaces on $\Omega$ intermediate between $L^1(\Omega)$ and $L^\infty(\Omega)$.

scholarly journals Semi-embeddings of Banach spaces which are hereditarily c0

1983 ◽  
Vol 26 (2) ◽  
pp. 163-167 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Closed Unit Ball ◽  
Continuous Linear ◽  
Scattered Space ◽  
Complete Set

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

The path functor and faithful representability of banach lie algebras

1973 ◽  
Vol 16 (1) ◽  
pp. 54-69 ◽  
Author(s):  
W. T. van Est ◽  
S. Świerczkowski
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Banach Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Uniform Norm

In this note “vector space” will mean “Banach space” unless otherwise specified. Accordingly “Lie algebra” will stand for “Banach Lie algebra”. Morphisms between Lie algebras will be assumed continuous. A Banach algebra B will be always assumed associative, and it will be also viewed as a Lie algebra with product [X, YXY− YX. In particular, the Lie algebra gl(V) of endomorphisms of a vector space V will be equipped with the uniform norm. A morphism of Lie algebras L → gl(V) will b called a representation of L in gl(V). Also, if B is a Banach algebra, a morphism of Lie algebras L → B will be called a representation of L in B. From such one evidently obtains a representation of L in gl(B). A representation will be called faithful if it is injective.

TWO NEW GENERALISED HYPERSTABILITY RESULTS FOR THE DRYGAS FUNCTIONAL EQUATION

2017 ◽  
Vol 95 (2) ◽  
pp. 269-280 ◽  
Author(s):  
LADDAWAN AIEMSOMBOON ◽  
WUTIPHOL SINTUNAVARAT
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Normed Space ◽  
Nonempty Subset ◽  
Image Position

Let $X$ be a nonempty subset of a normed space such that $0\notin X$ and $X$ is symmetric with respect to $0$ and let $Y$ be a Banach space. We study the generalised hyperstability of the Drygas functional equation $$\begin{eqnarray}f(x+y)+f(x-y)=2f(x)+f(y)+f(-y),\end{eqnarray}$$ where $f$ maps $X$ into $Y$ and $x,y\in X$ with $x+y,x-y\in X$. Our first main result improves the results of Piszczek and Szczawińska [‘Hyperstability of the Drygas functional equation’, J. Funct. Space Appl.2013 (2013), Article ID 912718, 4 pages]. Hyperstability results for the inhomogeneous Drygas functional equation can be derived from our results.

scholarly journals The evaluation functionals associated with an algebra of bounded operators

1969 ◽  
Vol 10 (1) ◽  
pp. 73-76 ◽  
Author(s):  
J. Duncan
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Finite Rank ◽  
Bounded Operators ◽  
Image Position

In this note we shall employ the notation of [1] without further mention. Thus X denotes a normed space and P the subset of X × X′ given byGiven a subalgebra of B(X), the set {Φ(X,f):(x,f) ∈ P} of evaluation functional on is denoted by II. We shall prove that if X is a Banach space and if contains all the bounded operators of finite rank, then Π is norm closed in ′. We give an example to show that Π need not be weak* closed in ″. We show also that FT need not be norm closed in ″ if X is not complete.

scholarly journals AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES

2013 ◽  
Vol 56 (2) ◽  
pp. 427-437 ◽  
Author(s):  
ANIL KUMAR KARN ◽  
DEBA PRASAD SINHA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Measure ◽  
Operator Ideal ◽  
The Other ◽  
Summable Sequence ◽  
Other Hand ◽  
Limited Operator

AbstractLet 1 ≤ p < ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.

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