scholarly journals The approximation problem for compact operators

1969 ◽  
Vol 1 (3) ◽  
pp. 397-401 ◽  
Author(s):  
S.R. Caradus
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Finite Rank ◽  
Reflexive Banach Space ◽  
Approximation Problem ◽  
Compact Operators ◽  
Sufficient Condition

The following sufficient condition is obtained for the uniform approximability of compact operators on a reflexive Banach space by operators of finite rank: if S is the unit ball of X and ø: X* → C(S) is the imbedding ø(x*)x = x*(x) then we require ø(X*) to be complemented in C(S).

scholarly journals Ideals of compact operators

2004 ◽  
Vol 77 (1) ◽  
pp. 91-110 ◽  
Author(s):  
Åsvald Lima ◽  
Eve Oja
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Approximation Property ◽  
Closed Subspace ◽  
Reflexive Banach Space ◽  
Compact Operators ◽  
Compact Approximation

AbstractWe give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

scholarly journals Some algebraic properties of F(X) and K(X)

1975 ◽  
Vol 19 (4) ◽  
pp. 353-361 ◽  
Author(s):  
Freda E. Alexander
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Approximation Property ◽  
Reflexive Banach Space ◽  
Compact Operators ◽  
Multiplication Operators ◽  
Weakly Compact ◽  
Dual Algebra ◽  
Left Multiplication ◽  
Algebraic Properties

Throughout we consider operators on a reflexive Banach space X. We consider certain algebraic properties of F(X), K(X) and B(X) with the general aim of examining their dependence on the possession by X of the approximation property. B(X) (resp. K(X)) denotes the algebra of all bounded (resp. compact) operators on X and F(X) denotes the closure in B(X) of its finite rank operators. The two questions we consider are:(1) Is K(X) equal to the set of all operators in B(X) whose right and left multiplication operators on F(X) (or on B(X)) are weakly compact?(2) Is F(X) a dual algebra?

Left approximate identities in algebras of compact operators on Banach spaces

1986 ◽  
Vol 104 (1-2) ◽  
pp. 169-175 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Compact Operators ◽  
The Other ◽  
Compact Set ◽  
Approximate Identities ◽  
Necessary And Sufficient

SynopsisWe study the existence of left approximate units, left approximate identities and bounded left approximate identities in the algebras (X)of all compact operators on a Banach space X and ℱ(X)− of all operators uniformly approximable by finite rank operators. In the case of bounded left approximate identities, necessary and sufficient conditions on X are obtained. In the other cases, sufficient conditions are obtained, together with an example of non-existence using a space constructed by Szankowski. The possibility of the sufficient conditions being also necessary depends on the question of whether every compact set is contained in the closure of the image of the unit ball under an operator in (X)(or ℱ(X)−). Sufficient conditions on X are obtained for this to be true, but it is conjectured that the answer for general X is negative.

scholarly journals A remark on a theorem of Caradus

1972 ◽  
Vol 6 (3) ◽  
pp. 355-356
Author(s):  
J.A. Johnson
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Weak Topology ◽  
Approximation Property ◽  
Approximation Problem

It is shown how a result of S.R. Caradus on the approximation problem can be obtained from known theorems.Terms used here are standard (see [1] or [3]).Let X denote a Banach space, S its unit ball in the weak topology, and X* the dual of X. In [1], the following theorem is proved: (I) If X is reflexive and X* (considered as a subspaoe of the continuous scalar-valued functions C(S) in the canonical way) is complemented in C(S), then X has the approximation property.It is our purpose to point out that (I) is a corollary to some known theorems that yield the stronger conclusion (II) below.

scholarly journals Some Results on Iterative Proximal Convergence and Chebyshev Center

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Laishram Shanjit ◽  
Yumnam Rohen ◽  
Sumit Chandok ◽  
M. Bina Devi
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Best Proximity Point ◽  
Normal Structure ◽  
Sufficient Condition ◽  
Chebyshev Center ◽  
Opial’S Condition ◽  
Relatively Nonexpansive

In this paper, we prove a sufficient condition that every nonempty closed convex bounded pair M , N in a reflexive Banach space B satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping T on M ∪ N satisfying T M ⊆ M and T N ⊆ N , to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping T on M ∪ N satisfying T N ⊆ N and T M ⊆ M , Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of N relative to M . Some illustrative examples are provided to support our results.

scholarly journals BLURRED MAXIMAL CYCLICALLY MONOTONE SETS AND BIPOTENTIALS

2010 ◽  
Vol 08 (04) ◽  
pp. 323-336 ◽  
Author(s):  
MARIUS BULIGA ◽  
GÉRY DE SAXCÉ ◽  
CLAUDE VALLÉE
Keyword(s):  
Banach Space ◽  
Differential Inclusion ◽  
Reflexive Banach Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Let X be a reflexive Banach space and Y its dual. In this paper, we find necessary and sufficient conditions for the existence of a bipotential for a blurred maximal cyclically monotone set. Equivalently, we find a necessary and sufficient condition on ϕ ∈ Γ0(X) so that the differential inclusion [Formula: see text] can be put in the form y ∈ ∂b(·, y)(x), with b a bipotential.

scholarly journals Spatiality of Derivations of Operator Algebras in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Quanyuan Chen ◽  
Xiaochun Fang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Left Ideal ◽  
Operator Algebras ◽  
Reflexive Banach Space ◽  
Compact Operators ◽  
Alternative Proof ◽  
Additive Constant ◽  
Minimal Left Ideal ◽  
The Ideal

Suppose thatAis a transitive subalgebra ofB(X)and its norm closureA¯contains a nonzero minimal left idealI. It is shown that ifδis a bounded reflexive transitive derivation fromAintoB(X), thenδis spatial and implemented uniquely; that is, there existsT∈B(X)such thatδ(A)=TA−ATfor eachA∈A, and the implementationTofδis unique only up to an additive constant. This extends a result of E. Kissin that “ifA¯contains the idealC(H)of all compact operators inB(H), then a bounded reflexive transitive derivation fromAintoB(H)is spatial and implemented uniquely.” in an algebraic direction and provides an alternative proof of it. It is also shown that a bounded reflexive transitive derivation fromAintoB(X)is spatial and implemented uniquely, ifXis a reflexive Banach space andA¯contains a nonzero minimal right idealI.

Stable models and reflexive banach spaces

1999 ◽  
Vol 64 (4) ◽  
pp. 1595-1600 ◽  
Author(s):  
José Iovino
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Reflexive Banach Space ◽  
Reflexive Banach Spaces ◽  
Stable Models

AbstractWe show that a formula φ(x, y) is stable if and only if φ is the pairing map on the unit ball of E × E*, where E is a reflexive Banach space. The result remains true if the formula φ is replaced by a set of formulas .

Approximate Identities in Banach Algebras of Compact Operators

1993 ◽  
Vol 36 (1) ◽  
pp. 45-53 ◽  
Author(s):  
Niels Grønbæk ◽  
George A. Willis
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
General Framework ◽  
Banach Algebras ◽  
Compact Operators ◽  
Approximate Identity ◽  
Approximation Properties ◽  
Approximate Identities ◽  
Set Up ◽  
Uniformly Closed

AbstractLet X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.

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